Algebra_Baldor.pdf

ALGEBRA
DR . AURELIO BALDOR
F~~dad~~, Di~ec~~~ ~ Jefe de ~a C~-
~ed~a de Ma~e~~~ica~ de~ C~~egi~
Baid~~, Haba~a, C~b~ .
Jefe de ~a C~~ed~a de Ma~e~~~ica~,
STEVENS ACADEMY, H~b~ke~,
Ne~-Je~~e~, U .S .A .
P~~fe~~~ de Ma~e~~~ica~, SAINT
PETER'S COLLEGE . Je~~e~ Ci~~,
Ne~-Je~~e~ .
CULTURAL CENTROAMERICANA, S . A .
CON GR~FICOS Y 6523
EJERCICIOS Y PROBLEMAS
CON RESPUESTAS
Ob~a a~~~bada ~ ~ec~~e~dada c~~~ ~e~~~ ~a~a
~~~ I~~~i~~~~~ de Seg~~da E~~e~a~~a de ~a Re-
~~b~ica ~~~ e~ Mi~i~~e~i~ de Ed~caci~~, ~~e~i~
i~f~~~e fa~~~ab~e de ~a J~~~a T~c~ica de Di-
~ec~~~e~ de I~~~i~~~~~ de Seg~~da E~~e~a~~a .
EDICION 1980
TOTALMENTE REVISADA POR EL AUTOR
De~~~i~~ Lega~ : M . 9 .747-1980
I . S . B . N . : 84-357-0062-3
EDICIONES Y DISTRIBUCIONES CODICE, S . A . MADRID

E~ ~~~~iedad i~~e~ec~~a~ .
Q~eda hech~ e~ de~~~i~~ ~~e ~~e~c~ibe ~a ~e~ ;
~~~hibida ~a ~e~~~d~cci~~ e~ ~~d~ ~ e~ ~a~~e .
I~~~e~~ ~~~ EDIME ORGANIZACION GRAFICA, S . A .
P~~~g~~~ I~d~~~~ia~ de A~~~~~~~~i~~~, ~~~ . 1
Ca~~e D ~~~ . 12
MOSTO LES (Mad~id)
I~~~e~~ e~ E~~a~a - P~i~~ed i~ S~ai~

Pa~a ~e~~~~de~ a ~a ge~~i~ defe~e~cia ~~e ha~ ~e~id~ c~~
e~~a ~b~a ~~~ P~~fe~~~e~ ~ A~~~~~~ de ~a A~~~ica La~i~a,
~ie~~~ i~~~~d~cid~, e~ ~a ~~e~e~~e edici~~, ~~a ~e~ie de ~e~~~a~
~~e ~ie~de~ a ~~e e~~e ~ib~~ ~ea ~~~ efica~ e i~~e~e~a~~e .
He~~~ ~~~c~~ad~ ~~e ~a ~~e~e~~aci~~ c~~~~i~~~a ~~~ ~~
~~~a ~~a ~~de~~~a f~e~~e de ~~~i~aci~~ ~a~a e~ ~~aba~~ e~c~-
~a~ . E~ c~~~e~id~ ha ~id~ c~idad~~a~e~~e ~e~i~ad~ ~ ~e ha~
i~~~~d~cid~ di~e~~~~ c~ad~~~ ~ ~ab~a~ ~a~a ~~ a~~e~di~a~e ~~~
~i~a~ ~ efec~i~~ . E~ ~~~ de~ c~~~~, e~ ~~ d~b~e a~~ec~~ e~~~~ic~
~ f~~ci~~a~, hace~ de e~~a ~b~a, ~i~ ~~ga~ a d~da~, e~ A~geb~a
~~~ ~edag~gica ~ ~~~ed~~a de ~a~ ~~b~icada~ ha~~a h~~ e~
idi~~a e~~a~~~ .
L~~ Edi~~~e~ ha~ e~~i~ad~ ~~~~~~~~ i~~~~d~ci~ a~g~~~~ a~a-
did~~ ~~e c~~~~ib~~a~ a c~~~~e~a~ e~ c~~~e~id~ de ~~~ ~~~g~a~a~
~ige~~e~ . Ta~e~ a~adid~~ ~~~, ~a~a e~~~e~a~ ~~~~ a~g~~~~, ~a~
N~~a~ ~~b~e e~ C~~ce~~~ de N~~e~~ ; N~~a ~~b~e ~a~ ca~~idade~
c~~~~e~a~ e i~agi~a~ia~ ~ e~ C~ad~~ de ~~~ Ti~~~ B~~ic~~ de
De~c~~~~~ici~~ Fac~~~ia~ .
E~~e~a~~~ ~~e e~ P~~fe~~~ad~ de Hi~~a~~a~~~ica ~e~a a~~i-
~a~a~ e~ i~ge~~e e~f~e~~~ ~e~did~ ~~~ ~~d~~ ~~~ ~~c~ic~~ ~~e
ha~ i~~e~~e~id~ e~ ~a c~~fecci~~ de e~~a ~b~a . S~~~ ~~~ ~~eda
~ei~e~a~ ~~e~~~~ ~~~ ~~~f~~d~ ag~adeci~ie~~~ ~~~ ~a ac~gida
~~e ~e ha~ di~~e~~ad~ ~ie~~~e .
L~~ EDITORES

C~~ ace~d~ada de~~ci~~ ~ ~~~~~ ~~g~~~~, dedic~ e~~e
e~f~e~~~ edi~~~ia~, a ~a i~~~~idab~e ~e~~~ia de ~i ~ad~e,
P~~fe~~~a D~~a A~a L~i~a Se~~a~~ ~ P~~ce~, ~~e f~e~a
P~e~ide~~a de e~~a E~~~e~a d~~a~~e ~~~ a~~~ 1921 a 1926 .
D~ . J~~~ A . L~~e~ Se~~a~~

CONCEPTO DE NUMERO EN LOS PUEBLOS PRIMI-
TIVOS (25,000-5,000 A . C .) Medi~ ~ c~~~a~ f~e~~~
~a~ ~~i~e~a~ ac~i~idade~ ~a~e~~~ica~ de~ h~~b~e ~~i-
~i~i~~ . Hacie~d~ ~a~ca~ e~ ~~~ ~~~~c~~ de ~~~ ~~b~~e~
~~g~aba~, e~~~~ ~~i~e~~~ ~~eb~~~, ~a ~edici~~ de~ ~ie~-
5
PRELIMINARES
O
O ~ ~LGEBRA e~ ~a ~a~a de ~a Ma~e~~~ica ~~e e~~~dia ~a ca~~idad c~~~i-
de~ada de~ ~~d~ ~~~ ge~e~a~ ~~~ib~e .
2 CAR~CTER DEL ALGEBRA Y SU DIFERENCIA
CON LA ARITMETICA
E~ c~~ce~~~ de ~a ca~~idad e~ A~geb~a e~ ~~ch~ ~~~ a~~~i~ ~~e e~
A~i~~~~ica .
E~ A~i~~~~ica ~a~ ca~~idade~ ~e ~e~~e~e~~a~ ~~~ ~~~e~~~ ~ ~~~~~ e~-
~~e~a~ ~a~~~e~ de~e~~i~ad~~ . A~~, 20 e~~~e~a ~~ ~~~~ ~a~~~ : ~ei~~e; ~a~a
e~~~e~a~ ~~ ~a~~~ ~a~~~ ~ ~e~~~ ~~e ~~~e hab~~ ~~e e~c~ibi~ ~~ ~~~e~~
di~~i~~~ de 20 .
E~ A~geb~a, ~a~a ~~g~a~ ~a ge~e~a~i~aci~~, ~a~ ca~~idade~ ~e ~e~~e~e~-
~a~ ~~~ ~edi~ de ~e~~a~, ~a~ c~a~e~ ~~ede~ ~e~~e~e~~a~ ~~d~~ ~~~ ~a~~~e~ .
A~~, a ~e~~e~e~~a e~ ~a~~~ ~~e ~~~~~~~~ ~e a~ig~e~~~, ~ ~~~ ~a~~~ ~~ede ~e-
~~e~e~~a~ 20 ~ ~~~ de 20 ~ ~e~~~ de 20, a ~~e~~~a e~ecci~~, a~~~~e c~~-
~ie~e ad~e~~i~ ~~e c~a~d~ e~ ~~ ~~~b~e~a a~ig~a~~~ a ~~a ~e~~a ~~ ~a~~~
de~e~~i~ad~, e~a ~e~~a ~~ ~~ede ~e~~e~e~~a~, e~ e~ ~i~~~ ~~~b~e~a, ~~~~
~a~~~ di~~i~~~ de~ ~~e ~e he~~~ a~ig~ad~ .
O NOTACION ALGEBRAICA
L~~ ~~~b~~~~ ~~ad~~ e~ A~geb~a ~a~a ~e~~e~e~~a~ ~a~ ca~~idade~ ~~~ ~~~
~~~e~~~ ~ ~a~ ~e~~a~ .
~~ ~ e~ ~~~~e~ de~ ~~~e~~ de a~i~a~e~ ~~e ~~~e~a~ ;
a~~ ~~~gi~ ~a A~i~~~~ica . E~ ~~ige~ de~ A~geb~a e~
~~~~e~i~~ . Pa~a~~~ cie~~~~ de ~ig~~~ ~a~a ~~e e~ h~~-
b~e a~ca~~a~a ~~ c~~ce~~~ ab~~~ac~~ de~ ~~~e~~, ba~e
i~di~~e~~ab~e ~a~a ~a f~~~aci~~ de ~a cie~cia a~geb~aica .

6 ~ ALGEBRA
L~~ ~~~e~~~ ~e e~~~ea~ ~a~a ~e~~e~e~~a~ ca~~idade~ c~~~cida~ ~ de-
~e~~i~ada~ .
La~ ~e~~a~ ~e e~~~ea~ ~a~a ~e~~e~e~~a~ ~~da c~a~e de ca~~idade~, ~a
~ea~ c~~~cida~ ~ de~c~~~cida~ .
La~ ca~~idade~ c~~~cida~ ~e e~~~e~a~ ~~~ ~a~ ~~i~e~a~ ~e~~a~ de~ a~fa-
be~~ : a, b, c, d . . .
La~ ca~~idade~ de~c~~~cida~ ~e ~e~~e~e~~a~ ~~~ ~a~ ~~~i~a~ ~e~~a~ de~
a~fabe~~ : ~, ~, ~, ~, ~, ~ .
U~a ~i~~a ~e~~a ~~ede ~e~~e~e~~a~ di~~i~~~~ ~a~~~e~ dife~e~ci~~d~~~~
~~~ ~edi~ de c~~i~~a~ ; ~~~ e~e~~~~ : a', a", a"', ~~e ~e ~ee~ a ~~i~a, a ~e-
g~~da, a ~e~ce~a, ~ ~a~bi~~ ~~~ ~edi~ de ~~b~~dice~ ; ~~~ e~e~~~~ : a ~, a2 ,
a 8 , ~~e ~e ~ee~ a ~~b~~~, a ~~bd~~, a ~~b~~e~ .
O FORMULAS
C~~~ec~e~cia de ~a ge~e~a~i~aci~~ ~~e i~~~ica ~a ~e~~e~e~~aci~~ de
~a~ ca~~idade~ ~~~ ~edi~ de ~e~~a~ ~~~ ~a~ f~~~~~a~ a~geb~aica~ .
F~~~~~a a~geb~aica e~ ~a ~e~~e~e~~aci~~, ~~~ ~edi~ de ~e~~a~, de ~~a
~eg~a ~ de ~~ ~~i~ci~i~ ge~e~a~ .
A~~, ~a Ge~~e~~~a e~~e~a ~~e e~ ~~ea de ~~ ~ec~~~g~~~ e~
A = b ~ h
ig~a~ a~ ~~~d~c~~ de ~~ ba~e ~~~ ~~ a~~~~a ; ~~eg~, ~~a~a~d~ A
a~ ~~ea de~ ~ec~~~g~~~, b a ~a ba~e ~ h a ~a a~~~~a, ~a f~~~~~a/
~e~~e~e~~a~~ de ~~ ~~d~ ge~e~a~' e~ ~~ea de
c~a~~~ie~ ~ec~~~g~~~, ~~e~ e~ ~~ea de ~~ ~ec-
~~~g~~~ dad~ ~e ~b~e~d~~ c~~ ~~~~ ~~~~i~~i~
A=b~h=3 ~.X2
b ~ h e~ ~a f~~~~~a a~~e~i~~ ~~~ ~~~ ~a~~~e~
.~2 ~ .=6 ~.2.
e~ e~ ca~~ dad~ . A~~, ~i ~a ba~e de ~~ ~ec-
~~~g~~~ e~ 3 ~ . ~ ~~ a~~~~a 2 ~ ., ~~ ~~ea ~e~~ :
E~ ~~ea de ~~~~ ~ec~~~g~~~ c~~a A = b ~ h =8 ~4~ 34 ~. = 28 ~ .2 . (1)
ba~e f~e~a 8 ~ . ~ ~~ a~~~~a 31 ~ . ~e~~a : /'
O SIGNOS DEL ALGEBRA
L~~ ~ig~~~ e~~~ead~~ e~ A~geb~a ~~~ de ~~e~ c~a~e~ : Sig~~~ de O~e-
~aci~~, Sig~~~ de Re~aci~~ ~ Sig~~~ de Ag~~~aci~~ .
O 6 SIGNOS DE OPERACION
E~ A~geb~a ~e ~e~ifica~ c~~ ~a~ ca~~idade~ ~a~ ~i~~a~ ~~e~aci~~e~ ~~e
e~ A~i~~~~ica : S~~a, Re~~a, M~~~i~~icaci~~, Di~i~i~~, E~~~aci~~ a P~~e~-
cia~ ~ E~~~acci~~ de Ra~ce~, ~~e ~e i~dica~ c~~ ~~~ ~ig~~~ ~ig~ie~~e~ :
E~ Sig~~ de ~a S~~a e~ +, ~~e ~e ~ee ~~~. A~~ a + b ~e ~ee "a ~~~ b" .
(I) E~ e~ Ca~ . XVIII, ~~gi~a 270, ~e e~~~dia a~~~ia~e~~e ~~d~ ~~ ~e~aci~~ad~ c~~ ~a~
f~~~~~a~ a~geb~aica~ .

~ ~ PRELIMINARES ~ 7
E~ Sig~~ de ~a Re~~a e~ -, ~~e ~e ~ee ~e~~~ . A~~, a- b ~e ~ee "a ~e-
~~~ b"
E~ Sig~~ de ~a M~~~i~~icaci~~ e~ ~, ~~e ~e ~ee ~~~~i~~icad~ ~~~ . A~~,
a ~ b ~e ~ee "a ~~~~i~~icad~ ~~~ b" .
E~ ~~ga~ de~ ~ig~~ ~ ~~e~e e~~~ea~~e ~~ ~~~~~ e~~~e ~~~ fac~~~e~ ~
~a~bi~~ ~e i~dica ~a ~~~~i~~icaci~~ c~~~ca~d~ ~~~ fac~~~e~ e~~~e ~a~~~~e~i~ .
A~~, a .b ~ (a)(b) e~~i~a~e~ a a ~ b .
E~~~e fac~~~e~ ~i~e~a~e~ ~ e~~~e ~~ fac~~~ ~~~~~ic~ ~ ~~~ ~i~e~a~ e~
~ig~~ de ~~~~i~~icaci~~ ~~e~e ~~i~i~~e . A~~ abc e~~i~a~e a a ~ b ~ c ; 5~~
e~~i~a~e a 5 ~ ~ ~ ~.
E~ Sig~~ de ~a Di~i~i~~ e~ -, ~~e ~e ~ee di~idid~ e~~~e . A~~, a - b ~e
~ee "a di~idid~ e~~~e b" . Ta~bi~~ ~e i~dica ~a di~i~i~~ ~e~a~a~d~ e~ di-
~ide~d~ ~ e~ di~i~~~ ~~~ ~~a ~a~a h~~i~~~~a~ . A~~, ~ e~~i~a~e a ~ -
. ~:
0
E~ Sig~~ de ~a E~e~aci~~ a P~~e~cia e~ e~ e~~~~e~~e,
~~e e~ ~~ ~~~e~~ ~e~~e~~ c~~~cad~ a~~iba ~ a ~a de- a 3 = aaa ; b 6 = bbbbb .
~echa de ~~a ca~~idad, e~ c~a~ i~dica ~a~ ~ece~ ~~e dicha
ca~~idad, ~~a~ada ba~e, ~e ~~~a c~~~ fac~~~ . A~~,
C~a~d~ ~~a ~e~~a ~~ ~ie~e e~~~~e~~e, ~~ e~~~~e~~e e~ ~a ~~idad .
A~~, a e~~i~a~e a a~ ; ~~~ e~~i~a~e a ~'~'~' .
E~ Sig~~ de Ra~~ e~ V, ~~a~ad~ ~ig~~ ~adica~, ~ ba~~ e~~e ~ig~~ ~e c~-
~~ca ~a ca~~idad a ~a c~a~ ~e ~e e~~~ae ~a ~a~~ . A~~, -, ,~a- e~~i~a~e a ~a~~ c~a-
d~ada de a, ~ ~ea, ~a ca~~idad ~~e e~e~ada a~ c~ad~ad~ ~e~~~d~ce ~a ca~-
~idad a ; e~~i~a~e a ~a~~ c~bica de b, ~ ~ea ~a ca~~idad ~~e e~e~ada
a~ c~b~ ~e~~~d~ce ~a ca~~idad b .
O 7 COEFICIENTE
E~ e~ ~~~d~c~~ de d~~ fac~~~e~, c~a~~~ie~a de ~~~ fac~~~e~ e~ ~~a~ad~
c~eficie~~e de~ ~~~~ fac~~~ .
A~~, e~ e~ ~~~d~c~~ 3a e~ fac~~~ 3 e~ c~eficie~~e de~ fac~~~ a e i~dica
~~e e~ fac~~~ a ~e ~~~a c~~~ ~~~a~d~ ~~e~ ~ece~, ~ ~ea 3a = a + a + a ; e~
e~ ~~~d~c~~ 5b, e~ fac~~~ 5 e~ c~eficie~~e de b e i~dica ~~e 5b=b+b-'-b+b+b .
E~~~~ ~~~ c~eficie~~e~ ~~~~~ic~~ .
E~ e~ ~~~d~c~~ ab, e~ fac~~~ a e~ c~eficie~~e de~ fac~~~ b, e i~dica ~~e
e~ fac~~~ b ~e ~~~a c~~~ ~~~a~d~ a ~ece~, ~ ~ea ab = b + b + b + b . . . a
~ece~ . E~~e e~ ~~ c~eficie~~e ~i~e~a~ .
E~ e~ ~~~d~c~~ de ~~~ de d~~ fac~~~e~, ~~~ ~ ~a~i~~ de e~~~~ ~~~ e~
c~eficie~~e de ~~~ ~e~~a~~e~ . A~~, e~ e~ ~~~d~c~~ abcd, a e~ e~ c~eficie~~e
de bcd ; ab e~ e~ c~eficie~~e de cd ; abc e~ e~ c~eficie~~e de d .
C~a~d~ ~~a ca~~idad ~~ ~ie~e c~eficie~~e ~~~~~ic~, ~~ c~eficie~~e
e~ ~a ~~idad . A~~, b e~~i~a~e a ~b ; abc e~~i~a~e a ~abc .

8 ~ ALGEBRA
8O SIGNOS DE RELACION
Se e~~~ea~ e~~~~ ~ig~~~ ~a~a i~dica~ ~a ~e~aci~~ ~~e e~i~~e e~~~e d~~
ca~~idade~ . L~~ ~~i~ci~a~e~ ~~~ :
=, ~~e ~e ~ee ig~a~ a . A~~, a = b ~e ~ee "a ig~a~ a b" .
>, ~~e ~e ~ee ~a~~~ ~~e . A~~, ~ + ~ > ~ ~e ~ee "~ + ~ ~a~~~ ~~e ~" .
O
<, ~~e ~e ~ee ~e~~~ ~~e . A~~, a < b + c ~e ~ee "a ~e~~~ ~~e b ~+ c" .
SIGNOS DE AGRUPACION
L~~ ~ig~~~ de ag~~~aci~~ ~~~ : e~ ~a~~~~e~i~ ~~di~a~i~ ( ), e~ ~a~~~~e-
~i~ a~g~~a~ ~ c~~che~e [ ], ~a~ ~~a~e~ ~~ ~ ~a ba~~a ~ ~~~c~~~
E~~~~ ~ig~~~ i~dica~ ~~e ~a ~~e~aci~~ c~~~cada e~~~e e~~~~ debe efec-
~~a~~e ~~i~e~~ . A~~, (a+ b)c i~dica ~~e e~ ~e~~~~ad~ de ~a ~~~a de a ~ b
debe ~~~~i~~ica~~e ~~~ c ; [a - b]~ i~dica ~~e ~a dife~e~cia e~~~e a ~ b debe
~~~~i~~ica~~e ~~~ ~ ; ~ a + b 1 _ ~ c - d ~ i~dica ~~e ~a ~~~a de a ~ b debe di-
~idi~~e e~~~e ~a dife~e~cia de c ~ d .
10 MODO DE RESOLVER LOS PROBLEMAS
EN ARITMETICA Y EN ALGEBRA
E~~~~e~~~ a c~~~i~~aci~~ ~~ e~e~~~~ ~a~a hace~ ~~~a~ ~a dife~e~cia
e~~~e e~ ~~~~d~ a~i~~~~ic~ ~ e~ a~geb~aic~ e~ ~a ~e~~~~ci~~ de ~~~b~e~a~,
f~~dad~ e~~e ~~~i~~ e~ ~a ~~~aci~~ a~geb~aica ~ e~ ~a ge~e~a~i~aci~~ ~~e
~~~a i~~~ica .
La~ edade~ de A ~ B ~~~a~ 48 a~~~ . Si ~a edad de B e~ 5 ~ece~ ~a
edad de A, ~~~~ edad ~ie~e cada ~~~?
METODO ARITMETICO
Edad de A ~~~ edad de B = 48 a~~~ .
C~~~ ~a edad de B e~ 5 ~ece~ ~a de A, ~e~d~e~~~ :
Edad de A ~~~ 5 ~ece~ ~a edad de A = 48 a~~~ .
METODO ALGEBRAICO
C~~~ ~a edad de A e~ ~~a ca~~idad de~c~~~cida ~a ~e~~e~e~~~ ~~~ ~.
Sea ~ =edad de A .
E~~~~ce~ 5~ =edad de B .
C~~~ a~ba~ edade~ ~~~a~ 48 a~~~, ~e~d~e~~~ :
~ + 5~ = 48 a~~~ ;
~ ~ea, 6~ = 48 a~~~ .
O ~ea,
111 eg~,
6 ~ece~ ~a edad de A = 48 a~~~ ;
Edad de A = 8 a~~~ . R .
Edad de B = 8 a~~~ ~ 5 = 40 a~~~ . R .

CANTIDADES POSITIVAS Y NEGATIVAS
Si 6 ~ece~ ~ e~~i~a~e a ~~ a~~~ . ~ ~a~d~~ ~a ~e~~a I~~~e (~e -1' a~~~,
~ ~ea ~ = 8 a~~~, edad de A . R .
E~~~~ce~ 5~ = 8 a~~~ ~ 5 = 40 a~~~, edad de B . R .
11 CANTIDADES POSITIVAS Y NEGATIVAS
E~ A~geb~a, c~a~d~ ~e e~~~dia~ ca~~idade~ ~~e ~~ede~ ~~~a~~e e~
d~~ ~e~~id~~ ~~~e~~~~ ~ ~~e ~~~ de c~~dici~~ ~ de ~~d~ de ~e~ ~~~e~~~~,
~e e~~~e~a e~ ~e~~id~, c~~dici~~ ~ ~~~d~ de ~e~ (~a~~~ ~e~a~i~~) de ~a ca~~i-
dad ~~~ ~edi~ de ~~~ ~ig~~~ + ~ -, a~~e~~~ie~d~ e~ ~ig~~ + a ~a~ ca~~ida-
de~ ~~~ada~ e~ ~~ ~e~~id~ de~e~~i~ad~ (ca~~idade~ ~~~i~i~a~) ~ a~~e~~~ie~-
d~ e~ ~ig~~ - a ~a~ ca~~idade~ ~~~ada~ e~ ~e~~id~ ~~~e~~~ a~ a~~e~i~~ (ca~-
~idade~ ~ega~i~a~) .
A~~, e~ habe~ ~e de~ig~a c~~ e~ ~ig~~ + ~ ~a~ de~da~ c~~ e~ ~ig~~ - .
Pa~a e~~~e~a~ ~~e ~~a ~e~~~~a ~ie~e $100 de habe~, di~e~~~ ~~e ~ie~e
+ $100, ~ ~a~a e~~~e~a~ ~~e debe $100, di~e~~~ ~~e ~ie~e - $100 .
L~~ g~ad~~ ~~b~e ce~~ de~ ~e~~~~e~~~ ~e de~ig~a~ c~~ e~ ~ig~~ + ~
~~~ g~ad~~ ba~~ ce~~ c~~ e~ ~ig~~ - . A~~, ~a~a i~dica~ ~~e e~ ~e~~~~e~~~
~a~ca 100 ~~b~e ce~~ e~c~ibi~e~~~ + 100 ~ ~a~a i~dica~ ~~e ~a~ca 8~ ba~~
ce~~ e~c~ibi~e~~~ -8~
E~ ca~i~~ ~ec~~~id~ a ~a de~echa ~ hacia a~~iba de ~~ ~~~~~ ~e de~ig-
~a c~~ e~ ~ig~~ + ~ e~ ca~i~~ ~ec~~~id~ a ~a i~~~ie~da ~ hacia aba~~ de
~~ ~~~~~ ~e ~e~~e~e~~a c~~ e~ ~ig~~ - . A~~, ~i he~~~ ~ec~~~id~ 200 ~ .
a ~a de~echa de ~~ ~~~~~ dad~, di~e~~~ ~~e he~~~ ~ec~~~id~ +200 ~ .,
~ ~i ~ec~~~e~~~ 300 ~ . a ~a i~~~ie~da de ~~ ~~~~~ e~c~ibi~e~~~ -300 ~ .
E~ ~ie~~~ ~~a~~c~~~id~ de~~~~~ de C~i~~~ ~e c~~~ide~a ~~~i~i~~ ~ e~
~ie~~~ ~~a~~c~~~id~ a~~e~ de C~i~~~, ~ega~i~~ . A~~, + 150 a~~~ ~ig~ifica
150 a~~~ D . C . ~ - 78 a~~~ ~ig~ifica 78 a~~~ A . C .
E~ ~~ ~~~~e i~~~~d~cid~ e~ e~ ~~e~~, ~e~~e~e~~a~~~ c~~ e~ ~ig~~ + ~a
~~~ci~~ ~~e ~e ha~~a de~ ~~e~~ hacia a~~iba ~ c~~ e~ ~ig~~ - ~a ~~~ci~~ ~~e
~e ha~~a de~ ~~e~~ hacia aba~~ . A~~, ~a~a e~~~e~a~ ~~e ~a ~~~gi~~d de~ ~~~-
~e ~~e ~e ha~~a de~ ~~e~~ hacia a~~iba ~ide 15 ~ ., e~c~ibi~e~~~ + 15 ~ .,
~ ~i ~a ~~~ci~~ i~~~~d~cida e~ e~ ~~e~~ e~ de 8 ~ ., e~c~ibi~e~~~ - 8 ~ .
La ~a~i~~d ~~~~e ~e de~ig~a c~~ e~ ~ig~~ + ~ ~a ~a~i~~d ~~~ c~~ e~ ~ig-
~~ - ; ~a ~~~gi~~d e~~e ~e c~~~ide~a ~~~i~i~a ~ ~a ~~~gi~~d ~e~~e, ~ega~i~a .
P~~ ~~ ~a~~~, ~~ ~~~~~ de ~a Tie~~a c~~a ~i~~aci~~ ge~g~~fica ~ea : + 45~
de ~~~gi~~d ~ -15~ de ~a~i~~d ~e ha~~a~~ a 45~ a~ e~~e de~ ~~i~e~ ~e~idia-
~~ ~ a 15~ ba~~ e~ Ec~ad~~ .
12 ELECCION DEL SENTIDO POSITIVO
La fi~aci~~ de~ ~e~~id~ ~~~i~i~~ e~ ca~~idade~ ~~e ~~ede~ ~~~a~~e e~
d~~ ~e~~id~~ ~~~e~~~~ e~ a~bi~~a~ia, de~e~de de ~~e~~~a ~~~~~~ad ; e~ deci~,
* 9

~~ ALGEBRA
~~e ~~de~~~ ~~~a~ c~~~ ~e~~id~ ~~~i~i~~ e~ ~~e ~~e~a~~~ ; ~e~~ ~~a ~e~
fi~ad~ e~ ~e~~id~ ~~~i~i~~, e~ ~e~~id~ ~~~e~~~ a ~~~e ~e~~ e~ ~ega~i~~ .
A~~, ~i ~~~a~~~ c~~~ ~e~~id~ ~~~i~i~~ e~ ca~i~~ ~ec~~~id~ a ~a de~e-
cha de ~~ ~~~~~, e~ ca~i~~ ~ec~~~id~ a ~a i~~~ie~da de e~e ~~~~~ ~e~~
~ega~i~~, ~e~~ ~ada ~~~ i~~ide ~~~a~ c~~~ ~~~i~i~~ e~ ca~i~~ ~ec~~~id~
a ~a i~~~ie~da de~ ~~~~~ ~ e~~~~ce~ e~ ca~i~~ ~ec~~~id~ a ~a de~echa de~
~~~~~ ~e~~a ~ega~i~~ .
A~~, ~i ~~b~e e~ ~eg~e~~~ AB ~~~a~~~ c~~~ ~~~i~i~~ e~ ~e~~id~ de A
hacia B, e~ ~e~~id~ de
B hacia A ~e~~a ~ega . + +
~i~~, ~e~~ ~i fi~a~~~
c~~~ ~e~~id~ ~~~i~i~~ A B A
de B hacia A, e~ ~e~~i-
d~ de A hacia B ~e~~a
~ega~i~~ .
N~ ~b~~a~~e, e~ ~a ~~~c~ica ~e ace~~a~ ge~e~a~~e~~e ~~~ ~e~~id~~ ~~~i-
~i~~~ de ~~e ~e ~~a~~ e~ e~ ~~~e~~ a~~e~i~~ .
13 CERO e~ ~a a~~e~cia de ca~~idad . A~~, ~e~~e~e~~a~ e~ e~~ad~ ec~~~~i-
c~ de ~~a ~e~~~~a ~~~ 0 e~~i~a~e a deci~ ~~e ~~ ~ie~e habe~ ~i de~da~ .
La~ ca~~idade~ ~~~i~i~a~ ~~~ ~a~~~e~ ~~e 0 ~ ~a~ ~ega~i~a~ ~e~~~e~
~~e 0 . A~~, + 3 e~ ~~a ca~~idad ~~e e~ ~~e~ ~~idade~ ~a~~~ ~~e 0 ; + 5 e~
~~a ca~~idad ~~e e~ ci~c~ ~~idade~ ~a~~~ ~~e 0, ~ie~~~a~ ~~e - 3 e~ ~~a
ca~~idad ~~e e~ ~~e~ ~~idade~ ~e~~~ ~~e 0 ~ - 5 e~ ~~a ca~~idad ~~e e~
ci~c~ ~~idade~ ~e~~~ ~~e 0 .
De d~~ ca~~idade~ ~~~i~i~a~, e~ ~a~~~ ~a de ~a~~~ ~a~~~ ab~~~~~~ ; a~~,
+ 5 e~ ~a~~~ ~~e + 3, ~ie~~~a~ ~~e de d~~ ca~~idade~ ~ega~i~a~ e~ ~a~~~
~a de ~e~~~ ~a~~~ ab~~~~~~ : - 3 e~ ~a~~~ ~~e - 5 ; - 9 e~ ~e~~~ ~~e - 4 .
EJERCICIOS SOBRE CANTIDADES POSITIVAS
Y NEGATIVAS
1) U~ h~~b~e c~b~a $130 . Paga ~~a de~da de $80 ~ ~~eg~ hace c~~-
~~a~ ~~~ ~a~~~ de $95 . ~C~~~~~ ~ie~e?
Te~ie~d~ $130, ~ag~ $80 ; ~~eg~, ~e ~~ed~ c~~ $50 . De~~~~~ hace ~~
ga~~~ de $95 ~ c~~~ ~~~~ ~ie~e $50 i~c~~~e e~ ~~a de~da de $45 . P~~ ~~
~a~~~, ~ie~e ac~~a~~e~~e - $45 . R .
IF EJERCICIO 1
1 . Ped~~ deb~a 60 b~~~~a~e~ ~ ~ecibi~ 320 . E~~~e~a~ ~~ e~~ad~ ec~~~~ic~ .
2 . U~ h~~b~e ~~e ~e~~a 1170 ~~c~e~ hi~~ ~~a c~~~~a ~~~ ~a~~~ de 1515 .
E~~~e~a~ ~~ e~~ad~ ec~~~~ic~ .
3 . Te~~a $200 . C~b~~ $56 ~ ~ag~~ de~da~ ~~~ $189 . ~C~~~~~ ~e~g~?
B

CANTIDADES POSITIVAS Y NEGATIVAS ~ 11
4 . C~~~~~ ~~~a~ ~~~ ~a~~~ de 665 ~~~e~ ~ a~i~e~~~~ ~~~ 1178 . Si de~~~~~
~ecib~ 2280, ~c~~~ e~ ~i e~~ad~ ec~~~~ic~?
5 . Te~~a $20 . Pag~~ $15 ~~e deb~a, de~~~~~ c~b~~ $40 ~ ~~eg~ hice ga~~~~
~~~ $75. ~C~~~~~ ~e~g~?
6 . E~~i~~e hace ~~a c~~~~a ~~~ $67 ; de~~~~~ ~ecibe $72 ; ~~eg~ hace ~~~a
c~~~~a ~~~ $1( ; ~ de~~~~~ ~ecibe $2 . E~~~e~a~ ~~ e~~ad~ ec~~~~ic~ .
7 . De~~~~~ de ~ecibi~ 200 c~~~~e~ hag~ ~~e~ ga~~~~ ~~~ 78, 81 ~ 93 . Recib~
e~~~~ce~ 41 ~ ~~eg~ hag~ ~~ ~~e~~ ga~~~ ~~~ 59 . ~C~~~~~ ~e~g~?
8 . Ped~~ ~e~~a ~~e~ de~da~ de $45, $66 ~ $79 ~e~~ec~i~a~e~~e . E~~~~ce~
~ecibe $200 ~ hace ~~ ga~~~ de $10 . ~C~~~~~ ~ie~e?
2) A ~a~ 6 a . ~. e~ ~e~~~~e~~~ ~a~ca - 40 . A ~a~ 9 a . ~ . ha ~~bid~
7~ ~ de~de e~~a h~~a ha~~a ~a~ 5 ~. ~ . ha ba~ad~ 11~ . E~~~e~a~ ~a ~e~~e-
~a~~~a a ~a~ 5 ~ . ~ .
A ~a~ 6 a. ~ . ~a~ca -4~ . C~~~ a ~a~ 9 a . ~ . ha ~~bid~ 7~, c~~~a~~~
~ie~e di~i~i~~e~ de ~a e~ca~a de~de -4~ hacia a~~iba ~ ~e~d~e~~~ 3~ ~~b~e
ce~~ (+3~) ; c~~~ de~de e~~a h~~a ha~~a ~a~ 5 ~ . ~i . ha ba~ad~ 11~, c~~~a~d~
11 di~i~i~~e~ de ~a e~ca~a de~de +3~ hacia aba~~ ~~ega~e~~~ a -8~ . L~e-
g~, a ~a~ 5 ~ . ~ . ~a ~e~~e~a~~~a e~ de -8~ . R .
. EJERCICIO 2
1 . A ~a~ 9 a . ~ . e~ ~e~~~~e~~~ ~a~ca +12~ ~ de e~~a h~~a a ~a~ 8 ~ . ~ . ha
ba~ad~ 15 ~. E~~~e~a~ ~a ~e~~e~a~~~a a ~a~ 8 ~ . ~ .
2 . A ~a~ 6 a . ~ . e~ ~e~~~~e~~~ ~a~ca -3~ . A ~a~ 10 a . ~ . ~a ~e~~e~a~~~a
e~ 8 ~ ~~~ a~~a ~ de~de e~~a h~~a ha~~a ~a~ 9 ~ . ~ . ha ba~ad~ 6~ . E~~~e~a~
~a ~e~~e~a~~~a a ~a~ 9 ~ . ~ .
3 . A~a 1 ~~ ~. e~ ~e~~~~e~~~ ~a~ca +15~ ~ a ~a~ 10 ~ . ~ . ~a~ca -3 0 .
~C~~~~~~ g~ad~~ ha ba~ad~ ~a ~e~~e~a~~~a?
4 . A ~a~ 3 a . ~ . e~ ~e~~~~e~~~ ~a~ca -8~ ~ a~ ~edi~d~a +5~ . ~C~~~~~~
g~ad~~ ha ~~bid~ ~a ~e~~e~a~~~a?
5 . A ~a~ 8 a . ~ . e~ ~e~~~~e~~~ ~a~ca -4~ ; a ~a~ 9 a . ~ . ha ~~bid~ 7~ ; a
~a~ 4 ~ . ~ . ha ~~bid~ 2~ ~~~ ~ a ~a~ 11 ~ . ~ . ha ba~ad~ 11 ~ . E~~~e~a~
~a ~e~~e~a~~~a a ~a~ 11 ~ . ~ .
6 . A ~a~ 6 a . i~ . e~ ~e~~~~e~~~ ~a~ca -8~ . De ~a~ 6 a . ~ . a ~a~ 11 a . ~ .
~~be a ~a~~~ de 4~ ~~~ h~~a . E~~~e~a~ ~a ~e~~e~a~~~a a ~a~ 7 a . ~ ., a
~a~ 8 a . ~ . ~ a ~a~ 11 a . ~ .
7 . A ~a~ 8 a . ~ . e~ ~e~~~~e~~~ ~a~ca -1~ . De ~a~ 8 a . ~ . a ~a~ 11 a . ~. ba~a
a ~a~~~ de 2~ ~~~ h~~a ~ de 11 a . ~ . a 2 ~ . M . ~~be a ~a~~~ de 3~ ~~~
h~~a . E~~~e~a~ ~a ~e~~e~a~~~a a ~a~ 10 a . ~ ., a ~a~ 11 a . ~ ., a ~a~ 12 a . ~ .
~ a ~a~ 2 ~ . ~ .
8 . E~ d~a 10 de dicie~b~e ~~ ba~c~ ~e ha~~a a 56~ a~ ~e~~e de~ ~~i~e~
~e~idia~~ . De~ d~a 10 a~ 18 ~ec~~~e 7~ hacia e~ e~~e . E~~~e~a~ ~~ ~~~-
gi~~d e~~e d~a .
9 . E~ d~a ~~i~e~~ de feb~e~~ ~a ~i~~aci~~ de ~~ ba~c~ e~ : 71~ de ~~~gi~~d
~e~~e ~ 15 ~ de ~a~i~~d ~~~ . De~ d~a ~~i~e~~ a~ 26 ha ~ec~~~id~ 5~ hacia
e~ e~~e ~ ~~ ~a~i~~d e~ e~~~~ce~ de 5 0 ~~~ a~ ~~~ . E~~~e~a~ ~~ ~i~~aci~~
e~ d~a 26 .

12 ~ ALGEBRA
10 . E~ d~a 5 de ~a~~ ~a ~i~~aci~~ de ~~ ~ia~e~~ e~ 18~ de ~~~gi~~d e~~e ~
65 ~ de ~a~i~~d ~~~~e . De~ d~a 5 a~ 31 ha ~ec~~~id~ 3~ hacia e~ e~~e ~ ~e
ha ace~cad~ 4~ a~ Ec~ad~~ . E~~~e~a~ ~~ ~i~~aci~~ e~ d~a 31 .
11 . U~a ci~dad f~~dada e~ a~~ 75 A . C . f~e de~~~~ida 135 a~~~ de~~~~~ .
E~~~e~a~ ~a fecha de ~~ de~~~~cci~~ .
3) U~ ~~~i~ ~ec~~~e 40 ~. e~ ~~~ea ~ec~a a ~a de~echa de ~~ ~~~-
~~ A ~ ~~eg~ ~e~~~cede e~ ~a ~i~~a di~ecci~~ a ~a~~~ de 15 ~ . ~~~ ~eg~~-
d~ . E~~~e~a~ a ~~~ di~~a~cia ~e ha~~a de~ ~~~~~ A a~ cab~ de~ 1~, 2~, 39
~ 4~ ~eg~~d~ .
E~ ~~~i~ ha ~ec~~~id~ 40 ~ . a ~a de~echa de~ ~~~~~ A ; ~~eg~, ~~ ~~-
~ici~~ e~ + 40 i~ ., ~~~a~d~ c~~~ ~~~i~i~~ e~ ~e~~id~ de i~~~ie~da a de~echa .
E~~~~ce~ e~~ie~a a ~~~e~~e de ~a de~echa hacia ~a i~~~ie~da (~e~~id~
~ega~i~~) a ~a~~~ de 15 i~ . ~~~ ~eg~~d~ ; ~~eg~, e~ e~ ~~i~e~ ~eg~~d~ ~e
ace~ca 15 ~. a~ ~~~~~ A ~ c~~~ e~~aba a 40 ~ . de e~e ~~~~~, ~e ha~~a a
40 - 15 = 25 ~. a ~a de~echa de A ; ~~eg~, ~~ ~~~ici~~ e~ + 25 ~ . R .
E~ e~ 29 ~eg~~d~ ~e ace~ca ~~~~~ 15 ~. a~ ~~~~~ A ; ~~eg~, ~e ha~~a~~
a 25 - 15 = 10 ~ . a ~a de~echa de A ; ~~ ~~~ici~~ ah~~a e~ + 10 ~ . R .
E~ e~ 3c~ . ~eg~~d~ ~ec~~~e ~~~~~ 15 i~ . hacia A, ~ c~~~ e~~aba a
10 ~ . a ~a de~echa de A, hab~~ ~~egad~ a~ ~~~~~ A (c~~ 10 ~i .) ~ ~ec~~~i-
d~ 5 ~ . a ~a i~~~ie~da de A, e~ deci~, 10 - 15 = - 5 ~ . S~ ~~~ici~~ ah~~a
e~ -5 ~. R .
E~ e~ 49 ~eg~~d~ ~ec~~~e ~~~~~ 15 ~. ~~~ hacia ~a i~~~ie~da ~ c~~~
~a e~~aba a 5 ~ . a ~a i~~~ie~da de A, ~e ha~~a~~ a~ cab~ de~ 4 ~
~ ~eg~~d~ a
20 ~ . a ~a i~~~ie~da de A, ~ ~ea - 5 -15 = - 20 ~ . ; ~~eg~, ~~ ~~~ici~~
ah~~a e~ - 20 ~ . R .
- EJERCICIO 3
(SENTIDO POSITIVO : DE IZQUIERDA A DERECHA Y DE ABAJO A ARRIBA) .
1 . E~~~e~a~ ~~e ~~ ~~~i~ ~e ha~~a a 32 ~. a ~a de~echa de~ ~~~~~ A ; a
16 ~. a ~a i~~~ie~da de A .
2 . E~~~e~a~ ~~e ~a ~a~~e de ~~ ~~~~e ~~e ~~b~e~a~e de~ ~~e~~ e~ 10 ~ . ~
~ie~e e~~e~~ad~~ 4 ~ .
3 . De~~~~~ de ca~i~a~ 50 ~i . a ~a de~echa de~ ~~~~~ A ~ec~~~~ 85 ~ . e~ ,
~e~~id~ c~~~~a~i~ . ~A ~~~ di~~a~cia ~e ha~~~ ah~~a de A?
4 . Si c~~~~ a ~a i~~~ie~da de~ ~~~~~ B a ~a~~~ de 6 ~ . ~~~ ~eg~~d~, ~a
~~~ di~~a~cia de B ~e ha~~a~~ a~ cab~ de 11 ~eg~ .?
5 . D~~ c~~~ed~~e~ ~a~~e~ de~ ~~~~~ A e~ ~e~~id~~ ~~~e~~~~ . E~ ~~e c~~~e
hacia ~a i~~~ie~da de A ~a a S ~ . ~~~ ~eg . ~ e~ ~~e c~~~e hacia ~a de~echa
~a a 9 ~i . ~~~ ~eg . E~~~e~a~ ~~~ di~~a~cia~ de~ ~~~~~ A a~ cab~ de 6 ~eg .
6 . Pa~~ie~d~ de ~a ~~~ea (~e ~a~ida hacia ~a de~echa ~~ c~~~ed~~ da d~~ ~~e~~a~
a ~~a ~i~~a de 400 ~ . de ~~~gi~~d . Si ~~ ~a~~~ de~ ~i~~~ ~~~~~ ~ d~~
3 ~~e~~a~ a ~a ~i~~a e~ ~e~~id~ c~~~~a~i~, ~~~~ di~~a~cia he~~~ ~ec~~~id~?
7 . U~ ~~~~e de 40 ~ie~ de ~~~gi~~d ~e~~a 15 ~ie~ ~~b~e e~ ~~e~~ . D~a~ de~~~~~
~e i~~~~d~~e~~~ 3 ~ie~ ~~~ . E~~~e~a~ ~a ~a~~e ~~e ~~b~e~a~e ~ ~a e~~e~~ada .

CANTIDADES POSITIVAS Y NEGATIVAS ~ 13
8 . U~ ~~~i~ ~ec~~~e 55 ~i . a ~a de~echa de~ ~~~~~ A ~ ~~eg~ e~ ~a ~i~~a
di~ecci~~ ~e~~~cede 52 ~i . ~A ~~~ di~~a~cia ~e ha~~a de A?
9 . U~ ~~~i~ ~ec~~~e 32 ~ . a ~a i~~~ie~da de~ ~~~~~ A ~ ~~eg~ ~e~~~cede
e~ ~a ~i~~a di~ecci~~ 15 ~ . ~A ~~~ di~~a~cia ~e ha~~a de A?
10 . U~ ~~~i~ ~ec~~~e 35 ~~i . a ~a de~echa de B ~ ~~eg~ ~e~~~cede e~ ~a ~i~~a
di~ecci~~ 47 ~i . ;A ~~~ di~~a~cia ~e Da~~a de B?
11 . U~ ~~~i~ ~ec~~~e 39 ~i . a ~a i~~~ie~da de A1 ~ ~~eg~ ~e~~~cede e~ ~a
~i~~a di~ecci~~ 56 ~ . ~A ~~~ di~~a~cia ~e ha~~a de M?
12 . A ~a~~i~ de~ ~~~~~ B ~~a ~e~~~~a ~ec~~~e 90 i~ . a ~a de~echa ~ ~e~~~-
cede, e~ ~a ~i~~a di~ecci~~, ~~i~e~~ 58 ~ . ~ ~~eg~ 36 ~ . ~A ~~~ di~~a~cia
~e ha~~a de B?
13 . U~ ~~~i~ ~ec~~~e 72 ~i . a ~a de~echa de A ~ e~~~~ce~ e~~ie~a a ~e~~~-
cede~ e~ ~a ~i~~a di~ecci~~, a ~a~~~ de 30 ~ . ~~~ ~eg . E~~~e~a~ ~~
di~~a~cia de~ ~~~~~ A a~ cab~ de~ 14, 24, 39 ~ 49 ~eg .
14 . U~ a~~~ ~ec~~~e 120 K~ . a ~a i~~~e~da de~ ~~~~~ M ~ ~~eg~ ~e~~~cede
a ~a~~~ e~e 60 K~~ . ~~~ h~~a . ~A ~~~ di~~a~cia ~e ha~~a de~ ~~~~~ M
a~ cab~ de ~a 1``, : ~ 4'' h~~a?
14 VALOR ABSOLUTO Y RELATIVO
Va~~~ ab~~~~~~ de ~~a ca~~idad e~ e~ ~~~e~~ ~~e ~e~~e~e~~a ~a ca~-
~idad ~~e~ci~die~d~ de~ ~ig~~ ~ ~e~~id~ de ~a ca~~idad, ~ ~a~~~ ~e~a~i~~ e~
e~ ~e~~id~ de ~a ca~~idad, ~e~~e~e~~ad~ ~~~ e~ ~ig~~ .
A~~, e~ ~a~~~ ab~~~~~~ de +$8 e~ $8, ~ e~ ~a~~~ ~e~a~i~~ habe~, e~~~e-
~ad~ ~~~ e~ ~ig~~ + ; e~ ~a~~~ ab~~~~~~ de -$20 e~ $20, ~ e~ ~a~~~ ~e~a~i~~
de~da, e~~~e~ad~ ~~~ e~ ~ig~~ - .
La~ ca~~idade~ +7~ ~ -7~ ~ie~e~ e~ ~i~~~ ~a~~~ ab~~~~~~, ~e~~ ~~
~a~~~ ~e~a~i~~ e~ ~~~e~~~, ~~e~ e~ ~~i~e~~ e~~~e~a g~ad~~ ~~b~e ce~~ ~ e~
~eg~~d~ ba~~ ce~~ ; -8 ~ -11 ~ie~e~ e~ ~i~~~ ~a~~~ ~e~a~i~~ (g~ad~~
ba~~ ce~~) ~ di~~i~~~ ~a~~~ ab~~~~~~ .
1?~ ~a~~~ ab~~~~~~ de ~~a ca~~idad a~geb~aica c~a~~~ie~a ~e ~e~~e~e~~a
c~~~ca~d~ e~ ~~~e~~ ~~e c~~~e~~~~da a dich~ ~a~~~ e~~~e d~~ ~~~ea~ ~e~-
~ica~e~ . A~~, e~ ~a~~~ ab~~~~~~ de + 8 ~e ~e~~e~e~~a 181 .
15 CANTIDADES ARITMETICAS Y ALGEBRAICAS
I)e ~~ e~~~e~~~ a~~e~i~~~e~~e ~e ded~ce ~a dife~e~cia e~~~e ca~~ida-
de~ a~i~~~~ica~ ~ a~geb~aica~ .
Ca~~idade~ a~i~~~~ica~ ~~~ ~a~ ~~e e~~~e~a~ ~~~a~e~~e e~ ~a~~~ ab~~-
~~~~ e~e ~a~ ca~~idade~ ~e~~e~e~~ad~ ~~~ ~~~ ~~~e~~~, ~e~~ ~~ ~~~ dice~ e~
~e~~id~ ~ ~a~~~ ~e~a~i~~ (~e ~a~ ca~~idade~ .
A~~, c~a~d~ e~ A~i~~~~ica e~c~ibi~~~ ~~e ~~a ~e~~~~a ~ie~e $5, ~e-
~e~~~ ~~~a~e~~e ~a idea de~ ~a~~~ ab~~~~~~ $5 de e~~a ca~~idad, ~e~~ c~~
e~~~ ~~ ~abe~~~ ~i ~a ~e~~~~a ~ie~e $5 de habe~ ~ de de~da . E~c~ibie~d~
~~e e~ ~e~~~~e~~~ ~a~ca 8~, ~~ ~abe~~~ ~i ~~~ ~~b~e ce~~ ~ ba~~ ce~~ .

14 ~ ALGEBRA
Ca~~idade~ a~geb~aica~ ~~~ ~a~ ~~e e~~~e~a~ e~ ~a~~~ ab~~~~~~ de ~a~
ca~~idade~ ~ ade~~~ ~~ ~e~~id~ ~ ~a~~~ ~e~a~i~~ ~~~ ~edi~ de~ ~ig~~ .
A~~, e~c~ibie~d~ ~~e ~~a ~e~~~~a ~ie~e +$5 e~~~e~a~~~ e~ ~a~~~ ab-
~~~~~~ $5 ~ e~ ~e~~id~ ~ ~a~~~ ~e~a~i~~ (habe~) e~~~e~ad~ ~~~ e~ ~ig~~ + ;
e~c~ibie~d~ -$8 e~~~e~a~~~ e~ ~a~~~ ab~~~~~~ $8 ~ e~ ~e~~id~ ~ ~a~~~ ~e~a-
~i~~ (de~da) e~~~e~ad~ ~~~ e~ ~ig~~ - ; e~c~ibie~d~ ~~e e~ ~e~~~~e~~~ ~a~-
ca +80 ~e~e~~~ e~ ~a~~~ ab~~~~~~ 8~ ~ e~ ~a~~~ ~e~a~i~~ (~~b~e ce~~) e~~~e-
~ad~ ~~~ e~ ~ig~~ +, ~ e~c~ibie~d~ -9~ ~e~e~~~ e~ ~a~~~ ab~~~~~~ 9~ ~ e~
~a~~~ ~e~a~i~~ (ba~~ ce~~) e~~~e~ad~ ~~~ e~ ~ig~~ - .
L~~ ~ig~~~ + ~ - ~ie~e~ e~ A~geb~a d~~ a~~icaci~~e~ : ~~a, i~dica~ ~a~
~~e~aci~~e~ de ~~~a ~ ~e~~a, ~ ~~~a, i~dica~ e~ ~e~~id~ ~ c~~dici~~ de ~a~
ca~~idade~ .
E~~a d~b~e a~~icaci~~ ~e di~~i~g~e ~~~~~e c~a~d~ ~~~ ~ig~~~ + ~ -
~ie~e~ ~a ~ig~ificaci~~ de ~~~a ~ ~e~~a, ~a~ e~~~e ~~~~i~~~ ~ e~~~e~i~~e~ i~-
c~~ida~ e~ ~a~~~~e~i~, c~~~ ~~~ e~e~~~~ e~ (+ 8) + (-4) ~ e~ (-7) - (+ 6) .
C~a~d~ ~a~ ~~ecedie~d~ a ~~ ~~~~i~~, ~a ~ea ~i~e~a~ ~ ~~~~~ic~, e~~~e~a~ e~
~e~~id~ ~~~i~i~~ ~ ~ega~i~~, c~~~ ~~~ e~e~~~~ e~ -a, + b, + 7, --- 8
~( REPRESENTACION GR~FICA DE LA SERIE
ALGEBRAICA DE LOS N~MEROS
Te~ie~d~ e~ c~e~~a ~~e e~ 0 e~ A~geb~a e~ ~a a~~e~cia de ~a ca~~i-
dad, ~~e ~a~ ca~~idade~ ~~~i~i~a~ ~~~ ~a~~~e~ ~~e 0 ~ ~a~ ~ega~i~a~ ~e~~-
~e~ ~~e 0, ~ ~~e ~a~ di~~a~cia~ ~edida~ hacia ~a de~echa ~ hacia a~~iba de
~~ ~~~~~ ~e c~~~ide~a~ ~~~i~i~a~ ~ hacia ~a i~~~ie~da ~ hacia aba~~ de ~~
~~~~~ ~ega~i~a~, ~a ~e~ie a~geb~aica de ~~~ ~~~e~~~ ~e ~~ede ~e~~e~e~~a~
de e~~e ~~d~ :
E~e~~~~~
-5 -4 -3 -2 -1 0 +1 +2 +3 4 5
NOMENCLATURA ALGEBRAICA
17 EXPRESION ALGEBRAICA e~ ~a ~e~~e~e~~aci~~ de ~~ ~~~b~~~ a~ge-
b~aic~ ~ de ~~a ~ ~~~ ~~e~aci~~e~ a~geb~aica~ .
a, 5~, / - 4~, (a+ b )c,
(5~ - 3~)a
~2 .
~g TERMINO e~ ~~a e~~~e~i~~ a~geb~aica ~~e c~~~~a de ~~ ~~~~ ~~~b~~~
~ de ~a~i~~ ~~~b~~~~ ~~ ~e~a~ad~~ e~~~e ~~ ~~~ e~ ~ig~~ + ~ - . A~~,
a, 3b, 2~~,
4a
- ~~~ ~~~~i~~~ .
3~

NOMENCLATURA ALGEBRAICA
~ 1 5
L~~ e~e~e~~~~ de ~~ ~~~~i~~ ~~~ c~a~~~ : e~ ~ig~~, e~ c~eficie~~e, ~a
~a~~e ~i~e~a~ ~ e~ g~ad~ .
P~~ e~ ~ig~~, ~~~ ~~~~i~~~ ~~~i~i~~~ ~~~ ~~e ~a~ ~~ecedid~~ de~ ~ig-
~~ + ~ ~ega~i~~~ ~~~ ~~e ~a~ ~~ecedid~~ de~ ~ig~~ - . A~~, + a, + 8~, + 9ab
~~~ ~~~~i~~~ ~~~i~i~~~ ~ - ~, - 5bc ~ -
b
~~~ ~~~~i~~~ ~ega~i~~~ .
E~ ~ig~~ + ~~e~e ~~i~i~~e de~a~~e de ~~~ ~~~~i~~~ ~~~i~i~~~ . A~~,
a e~~i~a~e a + a ; 3ab e~~i~a~e a + 3ab .
P~~ ~a~~~, c~a~d~ ~~ ~~~~i~~ ~~ ~a ~~ecedid~ de ~i~g~~ ~ig~~ e~
~~~i~i~~ .
E~ c~eficie~~e, c~~~ ~e di~~ a~~e~, e~ ~~~ c~a~~~ie~a, ge~e~a~~e~~e e~
~~i~e~~, de ~~~ fac~~~e~ de~ ~~~~i~~ . A~~, e~ e~ ~~~~i~~ 5a e~ c~eficie~~e
e~ 5 ; e~ - 3a 2 ~3' e~ c~eficie~~e e~ - 3 .
La ~a~~e ~i~e~a~ ~a c~~~~i~~~e~ ~a~ ~e~~a~ ~~e ha~a e~ e~ ~~~~i~~ . A~~,
3~ 3 ~ 4 ~8 ~ 4
e~ 5~~ ~a ~a~~e ~i~e~a~ e~ ~~ ; e~ 2ab ~a ~a~~e ~i~e~a~ e~
ab .
19 EL GRADO DE UN TERMINO ~~ede ~e~ de d~~ c~a~e~ : ab~~~~~~ ~ c~~
~e~aci~~ a ~~a ~e~~a .
G~ad~ ab~~~~~~ de ~~ ~~~~i~~ e~ ~a ~~~a de ~~~ e~~~~e~~e~ de ~~~
fac~~~e~ ~i~e~a~e~ . A~~, e~ ~~~~i~~ 4a e~ de ~~i~e~ g~ad~ ~~~~~e e~ e~~~-
~ie~~e de~ fac~~~ ~i~e~a~ a e~ 1 ; e~ ~~~~i~~ ab e~ de ~eg~~d~ g~ad~ ~~~~~e
~a ~~~a de ~~~ e~~~~e~~e~ de ~~~ fac~~~e~ ~i~e~a~e~ e~ 1 + 1 = 2 ; e~ ~~~~i~~
a 2 b e~ de ~e~ce~ g~ad~ ~~~~~e ~a ~~~a de ~~~ e~~~~e~~e~ de ~~~ fac~~~e~
~i~e~a~e~ e~ 2 + 1 = 3; 5a 4 b 3 c 2 e~ de ~~~e~~ g~ad~ ~~~~~e ~a ~~~a de ~~~ e~-
~~~e~~e~ de ~~~ fac~~~e~ ~i~e~a~e~ e~ 4 + 3 + 2 = 9 .
E~ g~ad~ de ~~ ~~~~i~~ c~~ ~e~aci~~ a ~~a ~e~~a e~ e~ e~~~~e~~e de
dicha ~e~~a . A~~ e~ ~~~~i~~ b~ 3 e~ de ~~i~e~ g~ad~ c~~ ~e~aci~~ a b ~ de
~e~ce~ g~ad~ c~~ ~e~aci~~ a ~ ; 4~ 2 ~4 e~ de ~eg~~d~ g~ad~ c~~ ~e~aci~~ a ~
~ de c~a~~~ g~ad~ c~~ ~e~aci~~ a ~ .
20 CLASES DE TERMINOS
T~~~i~~ e~~e~~ e~ e~ ~~e ~~ ~ie~e de~~~i~ad~~ ~i~e~a~ c~~i~ 5a,
6a4b3, 2a
5 3a
T~~~i~~ f~acci~~a~i~ e~ e~ ~~e ~ie~e de~~~i~ad~~ ~i~e~a~ c~~~ b .
T~~~i~~ ~aci~~a~ e~ e~ ~~e ~~ ~ie~e ~adica~, c~~~ ~~~ e~e~~~~~ a~~e-
~i~~e~, e i~~aci~~a~ e~ ~~e ~ie~e ~adica~, c~~~ ~ab,
3b
La
"T~~~i~~~ h~~~g~~e~~ ~~~ ~~~ ~~e ~ie~e~ e~ ~i~~~ g~ad~ ab~~~~~~ .
A~~, 4~''~ ~ 6~ 2 ~: ' ~~~ h~~~g~~e~~ ~~~~~e a~b~~ ~~~ de ~~i~~~ g~ad~
ab~~~~~~ .
T~~~i~~~ he~e~~g~~e~~ ~~~ ~~~ de di~~i~~~ g~ad~ ab~~~~~~, c~~~~ 5a,
~~e e~ de ~~i~e~ g~ad~, ~ 3a 2 , ~~e e~ de ~eg~~d~ g~ad~ .

16 ALGEBRA
If EJERCICIO 4
1 . Diga~e ~~~ c~a~e de ~~~~i~~~ ~~~ ~~~ ~ig~ie~~e~ a~e~die~d~ a~ ~ig~~, a
~i ~ie~e~ ~ ~~ de~~~i~ad~~ ~ a ~i ~ie~e~ ~ ~~ ~adica~ :
2a 5b 2 ~ 4a 2 b 3
5a 2 , - 4a 3 b, -, -
6
. ~, -C/5b2,
6
, -
2 . D~ga~e e~ g~ad~ ab~~~~~~ de ~~~ ~~~~i~~~ ~ig~ie~~e~ :
5a, -6a 2 b, a 2 b 2 , - 5a 3 b 4 C, 8~ 5 ~~, 4~ 2 ~3, - ~~~5
3 . D~ga~e e~ g~ad~ de ~~~ ~~~~i~~~ ~ig~ie~~e~ ~e~~ec~~ a cada ~~~ de ~~~
fac~~~e~ ~i~e~a~e~ :
-a 3 b 2 , -5~4 ~ 3, 6a 2 b~ 3 , - 4abc~ 2 , 10~ 2 ~3 b 4 c 5
4 . De ~~~ ~~~~i~~~ ~ig~ie~~e~ e~c~ge~ c~a~~~ ~~e ~ea~ h~~~g~~e~~ ~ ~~e~
he~e~~g~~e~~ :
-4a 3 b 2 , 6ab 3 , -~ 5 , 6~ 4 ~, -2a 3 ~4 , - ab 5 , 4abc~ 2 , - 2ac
5 . E~c~ibi~ ~~e~ ~~~~i~~~ e~~e~~~ ; d~~ f~acci~~a~i~~ ; d~~ ~~~i~i~~~, e~~e~~~ ~
~aci~~a~e~ ; ~~e~ ~ega~i~~~, f~acci~~a~i~~ e i~~aci~~a~e~ .
6 . E~c~ibi~ ~~ ~~~~i~~ de cada ~~~ de ~~~ g~ad~~ ab~~~~~~~ ~ig~ie~~e~ : de
~e~ce~ g~ad~, de ~~i~~~ g~ad~, de ~~d~ci~~ g~ad~, de d~ci~~ ~~i~~~
g~ad~, de ~ig~~i~~ g~ad~ .
7 . E~c~ibi~ ~~ ~~~~i~~ de d~~ fac~~~e~ ~i~e~a~e~ ~~e ~ea de c~a~~~ g~ad~ c~~
~e~aci~~ a ~a ~ ; ~~~~ de c~a~~~ fac~~~e~ ~i~e~a~e~ ~~e ~ea de ~~~~i~~
g~ad~ c~~ ~e~aci~~ a ~a ~ ; ~~~~ de ci~c~ fac~~~e~ ~i~e~a~e~ ~~e ~ea de
d~ci~~ g~ad~ c~~ ~e~aci~~ a ~a b .
CLASIFICACION DE LAS EXPRESIONES ALGEBRAICAS
21 MONOMIO e~ ~~a e~~~e~i~~ a~geb~aica
~~e c~~~~a de ~~ ~~~~ ~~~~i~~, c~~~-- -
22 POLINOMIO e~ ~~a e~~~e~i~~ a~geb~aica ~~e c~~~~a de ~~~ de ~~
~~~~i~~, c~~~~ a + b, a + ~ - ~, ~3 + 2~ 2 + ~ +7 .
a 2 5~~ 4
Bi~~~i~ e~ ~~ ~~~i~~~i~ ~~e a+b, ~-~, 3
6h 2
a 2
T~i~~~i~ e~ ~~ ~~~i~~~i~ ~~e a+b+c, ~2 -5~+6, 5~ 2 -6~ 3 + 3,
c~~~~a de ~~e~ ~~~~i~~~, c~~~
23 EL GRADO de ~~ ~~~i~~~i~ ~~ede ~e~ ab~~~~~~ ~ c~~ ~e~aci~~ a ~~a
~e~~a .
G~ad~ ab~~~~~~ de ~~ ~~~i~~~i~ e~ e~ g~ad~ de ~~ ~~~~i~~ de ~a~~~
g~ad~ . A~~, e~ e~ ~~~i~~~i~ ~ 4 - 5~ 3 + ~ 22 - 3~ e~ ~~i~e~ ~~~~i~~ e~ de
c~a~~~ g~ad~ ; e~ ~eg~~d~, de ~e~ce~ g~ad~ ; e~ ~e~ce~~, de ~eg~~d~ g~ad~, ~
e~ ~~~i~~, de ~~i~e~ g~ad~ ; ~~eg~, e~ g~ad~ ab~~~~~~ de~ ~~~i~~~i~ e~ e~
c~a~~~ .
c~~~~a de d~~ ~~~~i~~~, c~~~ :
~
2 ~
3a, - 5b, 43 .
a

NOMENCLATURA ALGEBRAICA ~ 1 7
G~ad~ de ~~ ~~~i~~~i~ c~~ ~e~aci~~ a ~~a ~e~~a e~ e~ ~a~~~ e~~~-
~e~~e de dicha ~e~~a e~ e~ ~~~i~~~i~ . A~~, e~ ~~~i~~~i~ a~ + a 4 ~2 -a 2 ~4 e~
de ~e~~~ g~ad~ c~~ ~e~aci~~ a ~a a ~ de c~a~~~ g~ad~ c~~ ~e~aci~~ a ~a ~ .
U~ ~~~i~~~i~ e~ e~~e~~ c~a~d~ ~i~g~~~ de ~~~ ~~~~i~~~ ~ie~e de~~-
2
~i~ad~~ ~i~e~a~ c~~~ ~ 2 + 5~ - 6 ;
2
-
3
+ 5
; f~acci~~a~i~ c~a~d~ a~g~~~
2
de ~~~ ~~~~i~~~ ~ie~e ~e~~a~ e~ e~ de~~~i~ad~~ c~~~
b
+ - 8 ; ~aci~~a~
c~a~d~ ~~ c~~~ie~e ~adica~e~, c~~~ e~ ~~~ e~e~~~~~ a~~e~i~~e~ ; i~~aci~~a~
c~a~d~ c~~~ie~e ~adica~, c~~~ V+--Ab_-~Z- VIa cb ; h~~~g~~e~ c~a~d~ ~~-
d~~ ~~~ ~~~~i~~~ ~~~ de~ ~i~~~ g~ad~ ab~~~~~~, c~~~ 4a 3 +5a 2 b+6ab 2 +b 3 ,
~ he~e~~g~~e~ c~a~d~ ~~~ ~~~~i~~~ ~~ ~~~ de~ ~i~~~ g~ad~, c~~~
~3 +~ 2 +~-6 .
P~~i~~~i~ c~~~~e~~ c~~ ~e~aci~~ a ~~a ~e~~a e~ e~ ~~e c~~~ie~e ~~d~~
~~~ e~~~~e~~e~ ~~ce~i~~~ de dicha ~e~~a, de~de e~ ~~~ a~~~ a~ ~~~ ba~~ ~~e
~e~ga dicha ~e~~a e~ e~ ~~~i~~~i~ . A~~, e~ ~~~i~~~i~ ~ 5 + ~4 - ~ 3 + ~2 - 3~
e~ c~~~~e~~ ~e~~ec~~ de ~a ~, ~~~~~e c~~~ie~e ~~d~~ ~~~ e~~~~e~~e~ ~~ce~i-
~~~ de ~a ~ de~de e~ ~~~ a~~~ 5, ha~~a e~ ~~~ ba~~ 1, ~ ~ea 5, 4, 3, 2, 1 ; e~
~~~i~~~i~ a 4 - a 3 b + a 2 b 2 - ab 3 + b 4 e~ c~~~~e~~ ~e~~ec~~ de a ~ b .
P~~i~~~i~ ~~de~ad~ c~~ ~e~~ec~~ a ~~a ~e~~a e~ ~~ ~~~i~~~i~ e~ e~
c~a~ ~~~ e~~~~e~~e~ de ~~a ~e~~a e~c~gida, ~~a~ada ~e~~a ~~de~a~~i~, ~a~
a~~e~~a~d~ ~ di~~i~~~e~d~ .
A~~, e~ ~~~i~~~i~ ~4 - 4~ 3 + 2~ 2 - 5~ + 8 e~~~ ~~de~ad~ e~ ~~de~ de~-
ce~de~~e c~~ ~e~aci~~ a ~a ~e~~a ~~de~a~~i~ ~ ; e~ ~~~i~~~i~ a 5 - 2a 4 b + 6a 3 b 2
-5a 2 b 8 + 3ab 4 - b 5 e~~~ ~~de~ad~ e~ ~~de~ de~ce~de~~e ~e~~ec~~ de ~a ~e~~a
~~de~a~~i~ a ~ e~ ~~de~ a~ce~de~~e ~e~~ec~~ de ~a ~e~~a ~~de~a~~i~ b .
25 O~de~a~ ~~ ~~~i~~~i~ e~ e~c~ibi~ ~~~ ~~~~i~~~ de ~~d~ ~~e ~~~ e~~~-
~e~~e~ de ~~a ~e~~a e~c~gida c~~~ ~e~~a ~~de~a~~i~ ~~ede~ e~ ~~de~ de~-
ce~de~~e ~ a~ce~de~~e . A~~, ~~de~a~ e~ ~~~i~~~i~ -5~ 8 +~ 5 -3~+~ 4 -~ 2 +6 e~
~~de~ de~ce~de~~e c~~ ~e~aci~~ a ~ ~e~~ e~c~ibi~ ~ 5 +~ 4 -5~ 3 -~ 2 -3~+6 .
O~de~a~ e~ ~~~i~~~i~ ~ 4 ~ - 7~ 2 ~ 3 - 5~ 5 + 6~~ 4 + ~5 - ~3 ~2 e~ ~~de~ a~-
ce~de~~e c~~ ~e~aci~~ a ~ ~e~~ e~c~ibi~~~ :
~ 5 +6~~ 4 --7~ 2 ~ 3 -~ 3 ~ 2 +~ 4 ~-5~ 5 .
W EJERCICIO 5
1 . D~ga~e e~ g~ad~ ab~~~~~~ de ~~~ ~ig~ie~~e~ ~~~i~~~i~~ :
a) ~ 3 +~ 2 +~ . c) a :'b-a 2 b 2 +ab 3 -b 4 .
b) 5a-3a 2 +4 .a 4 -6- d) ~ 5 -6~ 4 ~ 3 -4a 2 b+~ 2 ~ 4 -3~~ .
2 . D~ga~e e~ g~ad~ de ~~~ ~ig~ie~~e~ ~~~i~~~i~~ c~~ ~e~aci~~ a cada ~~a
24
de ~~~ ~e~~a~ :
a) a 3 +a 2 -ab 3 . c) 6a 4 b 7 -4a-~+ab 9 -5a&b~~~ .
b) ~ 4 +4~ 3 -6~ 2 ~4 -4~~ 5 . d) ~4~2-~~~+~~4~3-X8 + ~
15- ~ 11 .
CLASES DE POLINOMIOS

1 8 ~ ALGEBRA
26 T~~~i~~ i~de~e~die~~e de ~~ ~~~i~~~i~ c~~ ~e~aci~~ a ~~a ~e~~a e~
e~ ~~~~i~~ ~~e ~~ ~ie~e dicha ~e~~a .
A~~, e~ e~ ~~~i~~~i~ a 3 -a 2 +3a-5 e~ ~~~~i~~ i~de~e~die~~e c~~
~e~aci~~ a ~a a e~ 5 ~~~~~e ~~ ~ie~e a ; e~ ~4 - 6~ 3 + 8~ 2 - 9~ + 20 e~ ~~~~i-
~~ i~de~e~die~~e e~ 20 ; e~ a 3 - a 2 b + 3ab 2 + b 3 e~ ~~~~i~~ i~de~e~die~~e
c~~ ~e~aci~~ a ~a a e~ b 3 , ~ e~ ~~~~i~~ i~de~e~die~~e c~~ ~e~aci~~ a ~a b
e~ a3 . E~ ~~~~i~~ i~de~e~die~~e c~~ ~e~aci~~ a ~~a ~e~~a ~~ede c~~~ide~a~~e
~~e ~ie~e e~a ~e~~a c~~ e~~~~e~~e ce~~, ~~~~~e c~~~ ~e ~e~~ ~~~ ade~a~~e,
~~da ca~~idad e~e~ada a ce~~ e~~i~a~e a 1 .
A~~, e~ e~ ~~i~e~ e~e~~~~ a~~e~i~~, - 5 e~~i~a~e a - 5a~, ~ e~ e~ ~~~i-
~~ e~e~~~~, b 3 e~~i~a~e a a~b 3 .
N> EJERCICIO 6
1 . A~e~die~d~ a ~i ~ie~e~ ~ ~~ de~~~i~ad~~ ~i~e~a~ ~ a ~i ~ie~e~ ~ ~~ ~adi-
ca~, d~ga~e de ~~~ c~a~e ~~~ ~~~ ~~~i~~~i~~ ~ig~ie~~e~ :
a) a 3 +2a 2 -3a . c) a ~ V ~b_ - 2c +
3 2
b)
a4
-~ + ~ -a . d) da+
a
-6b+4 .
2 3 2 2
2 . E~c~ibi~ ~~ ~~~i~~~i~ de ~e~ce~ g~ad~ ab~~~~~~ ; de ~~i~~~ g~ad~ ab~~-
~~~~; de ~c~a~~ g~ad~ ab~~~~~~ : de deci~~~~i~~~ g~ad~ ab~~~~~~ .
3 . E~c~ibi~ ~~ ~~i~~~i~ de ~eg~~d~ g~ad~ ~e~~ec~~ de ~a ~ ; ~~ ~~~i~~~i~
c) X 4 ~- ~3 ~ 2 +~ 2 ~ 3 -~ 4 .
d~ga~e c~~~e~ ~~~ c~~~~e~~~ ~ ~e~~ec~~ de c~~~e~ ~e~~a~ .
6 . E~c~ibi~ ~~e~ ~~~i~~~i~~ h~~~g~~e~~ de ~e~ce~ g~ad~ ab~~~~~~ ; c~a~~~
de ~~i~~~ g~ad~ ab~~~~~~ ; d~~ ~~~i~~~i~~ c~~~~e~~~ .
7 . O~de~a~ ~~~ ~ig~ie~~e~ ~~~i~~~i~~ ~e~~ec~~ de c~a~~~ie~ ~e~~a e~ ~~de~
de~ce~de~~e :
a) ~2 +6~-~ 3 +~ 4 .
b) 6a~ 2 -5a 3 +2a 2 ~+~ 3 .
c) -a 2 b 3 +a 4 b+a 3 b 2 -ab 4 .
d) a 4 -5a+6a 3 -9a 2 +6 .
e) -~ 8 ~2 +~ 10 +3~ 4 ~~-~~~ 4
+~ 2 ~8 .
f) -3~~ 1 J~ 2 +4~~ 2 ~3 -8~~~ -10~ 3 ~G+~ 7 -7~O~ 4 +~ ~~~ .
8 . O~de~a~ ~~~ ~ig~ie~~e~ ~~~i~~~i~~ ~e~~ec~~ de c~a~~~ie~ ~e~~a e~ ~~de~
a~ce~de~~e :
a) a 2 -5a 3 +6a . d) a 2 b 4 +a 4 b 3 -a~b 2 +a ."b+b 5 .
b) ~-5~ 3 +6~ 2 +9~ 4 . e) ~ 12 -~~~~+~' 2 ~ 4 -~ 3 ~ ~~.
c) 2~'+4~ 5 -O~+2~ 2 +5~ 3 .
de ~~i~~~ g~ad~ ~e~~ec~~ de ~a a ; ~~ ~~~i~~~i~ de ~~~e~~ g~ad~ ~e~-
~ec~~ de ~a ~.
4 . De ~~~
a)
~ig~ie~~e~ ~~~i~~~i~~ :
3a 2 b+4a 3 -5b 3 . d) 4a-5b+6c 2 -8d 3 -6 .
b)
c)
a 4 -a 3 b+a 2 b 2 +ab 3 .
~-b~a+ab~ 3 +ab 3 ~2 .
e)
f)
~ 5 -a~a+a 2 ~ 3 -a 3 ~`-a 4 ~+~ 5 .
-6a 3 b 4 -5a~b+8a 2 b 5 -b 7 .
e~c~ge~ d~~ ~~e ~ea~ h~~~g~~e~~ ~ d~~ he~e~~g~~e~~ .
5 . De ~~~
a)
b)
~ig~ie~~e~ ~~~i~~~i~~ :
a 4 -a 2 +a-a 3 .
5~ 4 -8~ 2 +~-6 .
d) ~5 -~`+~0-~+5 .
e) ~5-b~'+b 2 ~ 3 -b 3 ~ 2 +b 4 ~
3

27 TERMINOS SEMEJANTES
D~~ ~ ~~~ ~~~~i~~~ ~~~ ~e~e~a~~e~ c~a~d~ ~ie~e~ ~a ~i~~a ~a~~e ~i~e-
~a~, ~ ~ea, c~a~d~ ~ie~e~ ig~a~e~ ~e~~a~ afec~ada~ de ig~a~e~ e~~~~e~~e~ .
E~e~~~~~ 2a ~ a ; - 2b ~ 8b ; - 5a 3 b 2 ~ - 8a 8 b 2 ; ~~+1 ~
3~~1+i .
L~~ ~~~~i~~~ 4ab ~ - 6a 2 b ~~ ~~~ ~e~e~a~~e~, ~~~~~e a~~~~e ~ie~e~
ig~a~e~ ~e~~a~, ~~~a~ ~~ ~ie~e~ ~~~ ~i~~~~ e~~~~e~~e~, ~a ~~e ~a a de~ ~~i-
~e~~ ~ie~e de e~~~~e~~e 1 ~ ~a a de~ ~eg~~d~ ~ie~e de e~~~~e~~e 2 .
L~~ ~~~~i~~~ - b~ 4 ~ ab 4 ~~ ~~~ ~e~e~a~~e~, ~~~~~e a~~~~e ~ie~e~ ~~~
~i~~~~ e~~~~e~~e~, ~a~ ~e~~a~ ~~ ~~~ ig~a~e~ .
28 REDUCCION DE TERMINOS SEMEJANTES e~ ~~a ~~e~aci~~ ~~e ~ie-
~e ~~~ ~b~e~~ c~~~e~~i~ e~ ~~ ~~~~ ~~~~i~~ d~~ ~ ~~~ ~~~~i~~~ ~e-
~e~a~~e~ .
E~ ~a ~ed~cci~~ de ~~~~i~~~ ~e~e~a~~e~ ~~ede~ ~c~~~i~ ~~~ ~~e~ ca~~~
~ig~ie~~e~ :
1) Red~cci~~ de d~~ ~ ~~~ ~~~~i~~~ ~e~e~a~~e~ de~ ~i~~~ ~ig~~ .
REGLA
Se ~~~a~ ~~~ c~eficie~~e~, ~~~ie~d~ de~a~~e de e~~a ~~~a e~ ~i~~~
~ig~~ ~~e ~ie~e~ ~~d~~ ~ a c~~~i~~aci~~ ~e e~c~ibe ~a ~a~~e ~i~e~a~ .
E~e~~~~~
(1) 3a + 2a = 5a . R . (6) 2ab + 3ab = eab . R .
(2) -5b-7b = -12b . R .
1 2
(7) -~~~-3~~ = -~~ . R .
(3) - a 2 -9a 2 =-100 2 . R . (8) 5~+~+2~=8~ . R .
(4) 3a~ -2 +5a~ - 2 =8a' -' R . (9) - ~-3~ - 6~ - 5~ = -15~ .
(5) - 4a~+ 1 -7a~' 1 =- 11a ~+1 . R . (10) ~~4~+~~ 2 ~+~~ 2 ~=?~ 2 ~. R .
2 4 8 8
. EJERCICIO 7
Red~ci~ :
1 . ~+2~ . 6 . -9~-7~ . 11-
2 . 8a+9a . 7 . 4a , +5a~ .
3 . 11 b+9b . 8 . 6a~ + 1+8a~ + 1 .
12-
4 . -b-5b . 9 . -~~+ 1 -5~~+ 1 .
5 . -8~-~ . 10 . -3a i-2-a~-2 . 13 .
REDUCCION DE TERMINO$ SEMEJANTES
1 1
2 a+ 2 Q .
3 ab+ 1 ab .
1 ~~+ 8
~~.
14 .
~ 19
1 4
- 5 ~~ -5 ~~.
15 . - 6
- 5 a 2 b -
8 1
a 2 b .
16 . -a--' ~ a .
R .

2) Red~cci~~ de d~~ ~~~~i~~~ ~e~e~a~~e~ de di~~i~~~ ~ig~~ .
REGLA
Se ~e~~a~ ~~~ c~eficie~~e~, ~~~ie~d~ de~a~~e de e~~a dife~e~cia e~ ~ig~~
de~ ~a~~~ ~ a c~~~i~~aci~~ ~e e~c~ibe ~a ~a~~e ~i~e~a~ .
E~e~~~~~
(1) 2a-3a=-a . R .
(2) 18~ - 11 ~ = 7~ . R .
(5) 25a~+ 1 -54a X ` 1 = - 29a~` 1 .
1 2 1
(6) 2a - ~a = --a . R .
R .
(3) - 20ab + 11 ab = -gab .
(4) - 8aX + 13a~ = 5aX . R .
R . (7) - 3a2b +02 b =
4a2 b. R .
~ ~
R .
c
(8) - 8 0 X+1 +~ a ~+1 =- Z ~*1 .
e
De ~a ~eg~a a~~e~i~~ ~e ded~ce ~~e d~~ ~~~~i~~~ ~e~e~a~~e~ de ig~a~e~ c~efi-
cie~~~~ ~ de ~ig~~ c~~~~a~i~ ~e a~~~a~ .
A~~ : - 8ab + B~b = 0 .
2 2
5
X 2 ~ - 5 X 2 ~=0 .
R .
R .
f EJERCICIO 8
Red~ci~ :
1 . 8a-6a . 5 . 2a-2a . 9 . 40~ 3 ~-51~ .3 ~ .
2 . 6a-8a . 6 . -7b+7b . 10 . -~ 2 ~+6~ 2 ~.
3 . 9ab-15ab . 7 . -14~~+32~~ . 11 . - 15~~+40~~ .
4 . 15ab-9ab . 8 . -25~ 2 ~+3'2~ 2 ~. 12 . 5500-810b 2 .
20 a ALGEBRA
17 . 8a+9a+6a . 29 . -~ 2 ~-8~ 2 ~-9~ 2 ~-20~ 2 ~.
18 . 15~+20~+~ . 30 . -3a~-5a~-6a"'-9a~ .
19 . -7~-8~-9~ . 31 . ~a + 9a+~a+a .
20 . -a~b-a~b-3a 2 b .
32 . ~a~+ 1 a~+ 1 a~ .
~~a~+
21 . a~+3a~+8a ~ .
22 . -5a~ + 1-3a~ + 1 -5a~ + 1 .
33 . 0 .5~+0 .6~+0 .7~+0 .8~ .
1 2 34 . - 1 ab- 1- ab--- ~- ab-ab .
23 . a+ 2 a+ ~a . 7 14 28
2 1 35 . -
2
~3~-
1
~3~-
1
~3 ~ - 12~ 3 ~ .
24 . -~--~- -~ .
3 6
36 . ab 2 +ab 2 + 7ab 2 +9ab 2 +21ab2 .
25 . 6 a~+ ~a~+a~ . 37 . -~~-~-8~-77~-3~ .
38 . -~a +- 1-8~a + 1-4~a + . 1-5~a + 1-~a + 1
26 . - a a 2 ~- 6 -a 2 ~-a 2 ~. 1 1
4 6 39 . 1
-~
-Z a+ a+ a+ a+ a .
27 . 11a+8a+9a+11a .
28 . ~~+~+3~~" 1 +4~~~ +1 +6,~ " 1 . 40 . - 1 ab--'ab- ab- ab- ab .
3 (1 2 12 9

UDUCCION DR TERMINOS UUMUJANTIS
3) Red~cci~~ de ~~~ de d~~ ~~~~i~~~ ~e~e~a~~e~ de ~ig~~~ di~~i~~~~ .
REGLA
Se ~ed~ce~ a ~~ ~~~~ ~~~~i~~ ~~d~~ ~~~ ~~~i~i~~~, ~e ~ed~ce~ a ~~ ~~~~
~~~~i~~ ~~d~~ ~~~ ~ega~i~~~ ~ a ~~~ d~~ ~e~~~~ad~~ ~b~e~id~~ ~e a~~ica ~a ~e-
g~a de~ ca~~ a~~e~i~~ .
40 21
13 . -~ 2 ~+~ 2 ~. 23 . - 4 ~=~+ 9 ~-~ . 33 . _~a+~+~ai 1 .
14 . -9ab 2 +9ab 2 . 3 5 34 . - 1 a~-2+ 1 a~-
15 . 7~ 2 ~-7~ 2 ~. 24 . -a~ - -a~~ .
K 4
4 -
6
a ~, +1_ 7 a~~, +1 .
16 . -101~~~+118~~ . 3 35 .
S5 . -a~a + -a~ . 6 12
17 . 502ab-405ab . ~
18 . -1024~+1018~ . 26 .
5 7
-~~ - -~~ . 36 . 4a2- 1 a 2 .
3
19 . -15ab+15ab .
a
27 . -a2b+ ~
a 2 b . 37 . -5~~+ 4
~
~~.
20 . 1 a-
~
a . 11
2 4 28 . 3 .4a'b 3 -5 .6a'b 3 . 38 . Sa~+ 2 b~ +3-25 a ~+2b~+3 .
8 1 29 . -1 .2~~+3 .41' : .
21 . 4-a
-
2
-a . 30 .
31 .
4a~-2a~ .
-Sa~ -'+~a~' 1 .
39 . _ 7 a~b~+a a'b ~ .
S
0 .85~~~ -' ~~~.
22 .
c
a2b-
~
a 2 b . 32 . 25~ ^ -- ' -32~~ -1 . 40 .
6 12
E~e~~~~~
(1) Red~ci~ 5a - 8a + a - 6a + 21~ .
Red~cie~d~ ~~~ ~~~i~i~~~ : 5a + a + 21a = 27a .
Red~cie~d~ ~~~ ~ega~i~~~ : - 8a - 6a = - 14a .
A~~ica~d~ a e~~~~ ~e~~~~ad~~ ~b~e~id~~, 27a ~ - 14a, ~a ~eg~a de~ ca~~ a~~e-
~i~~, ~e ~ie~e : 27a - 14a = 13a . R .
E~~a ~ed~cci~~ ~a~bi~~ ~~e~e hace~~e ~~~~i~~ a ~~~~i~~, de e~~a ~a~e~a :
5~-8a=-3a ; -3a+a=-2a ; -2~-6a=-8a ; -8a+21a=13a . R .
(2) Red~ci~ - b~ 2 + ~ b~' 2 + ~b~ 2 - 4b~_ + b~ .
Red~cie~d~ ~~~ ~~~i~i~~~ : ~ b~= + b~ 2 + b~ 2 =Z~b~ 2 .
22
Red~cie~d~ ~~~ ~ega~i~~~ : - ; b~ 2 - 4b~ 2 = - ~b~ 2 .
Te~d~e~~~ : -b~~ - ~
2 b~ 2 = - 20 b~ 2 . R .
20
M . EJERCICIO 9
Red~ci~ :
1 . 9a-3a+5a . 5 . 19~~-~O~+G~ . 1
2 . -8~+9~-~ . 6 . -~~ab-15ab+26ab . 9 . 3 ~+ ~-~ .
3 . 12~~-23~~-5~~ . 7 . - .ia~ +9a~-35a~ . 3 1 1
4 . -~+19~-18~ . 8 . -24a~+ 2 -15a~ F2+39a~+2 . 10 . --~ + -~ - -~ .
5 4 2

33 . _a~+ 1 +7a~+ 1 -11a~+ 1 -20a~+ 1 +26a ~ +' .
34 . a+6a-20a+150a-80a+31a .
35 . -9b-11b-17b-81b-b+110b .
36 . -a~b+15a 2 b+a~b-85a 2 b-131a 2 b+39a 2 b .
37 . 84~ 2 ~-501~ 2 ~-604~ 2 ~-715~-~+231~ 2 ~+165~-~ .
38 .
5 a362+2 a3b2-1a3b2- 5
-a~b 2 +4a 3 b 2.
8 3 4 ~8
39 . 40a-81a+130a+41a-83a-91a+16a .
40 . -21ab+52ab-60ab+84ab-31ab-ab-23ab .
29 REDUCCION DE UN POLINOMIO QUE CONTENGA TERMINOS
SEMEJANTES DE DIVERSAS CLASES
E~e~~~~~
( 1) Red~ci~ e~ ~~~i~~~i~ Sa - 6b + 8c + 9a - 20c - b + 6b - c .
Se ~ed~ce~ ~~~ ~e~a~ad~ ~~~ de cada c~a~e :
S~ + 9a = 14a .
-6b-b+6b=-b .
8c-20c-c=-13c .
Te~d~e~~~ : 14a - b -13c . R .
(2) Red~ci~ e~ ~~~i~~~i~ :
8a 3 b 2 + 4a 4 b 3 + 6a 8 b 2 - a 3 b 2 - 9a 4 b 3 - 15 - 5ab 5 + 8- 6ab 5 .
Se ~ed~ce~ ~~~ ~e~a~ad~ ~~~ de cada c~a~e : 4a 4 b 3 - 9a 4 b 3 = - 5a 4 b 8 .
8a 3 b2 + 6a&b 2 - a 3 b 2 = 13a 3 b 2 .
- 5ab 5 - 6ab 5 = - 11 ab 5 .
-15+8=- 7 .
Te~d~e~~~ : - 5a 4 b 8 + 13a 8 b 2 - 11 ab 5 - 7 . R .
(3) Red~ci~ e~ ~~~i~~~i~ :
~~4 -2~ 8 ~+3~ 4 -~ 4 +e~ 4 -0 .3~ 4 --~ 8 ~-6+~ 8 ~-14+2a~ 4 .
51
22 ALGEBRA
11 . ~2b+-'a2~-a~b . 23 . 2 b- 2 b+aea- 1$ a 2 b-a 2 b .
12 . -a+8a+9a-15a .
13 . ~ab-f~ab+20ab-31ab . 24 . - ~b2- e ab2+ab2- ~ab 2 .
14 . 25~ 2 -50~ 2 +11~ 2 +14~ 2 . 25 . -a+8a-11a+15a-75a .
15 . -~~-8~~-19~~+40~~ . 26 . -7c4+21c+14c-30c+82c .
16 . ~ab+2~ab-ab-80ab . 27 . -~~+~4~~-31~~-~~+20~~ .
17 . -25~~ 2 +~ ~~~ 2 +60~~ 2 -82~~ 2 . 28 . a 2 ~-7a 2 ~-93a 2 ~+51a 2 ~+48a 2 ~.
18 . -72a~+87a~-101 a~+243a~ . 29 . -a+a-a+a-3a+6a .
19 . -82b~-71b~-53b~+206b~ .
20 . 1050-4640+58a 3 +301a 8 . 30 .
1
~+ ~~- ~~+ Z ~-~ .
1 1 1 1
21 . ~- ~+ ~- ~ .
~ 3 4 5 31 . -2~+ 4~+ 4 ~+~- ~~ .
22 . 2~-~+ 1
- 12 ~. 32 . 7a ~- 30a ~-41a~-9a ~+73a~ .

Te~d~e~~~ :
6~4 + 3~ 4 - 0 .3~ 4 = 3 11X 4 .
VALOR NUMERICO
1 3 _ 1
~3~
2 X3 ~ 5X3~ 10
~3 ~ .
23~4+
6
~
4 -~4 -26~ 4 .
-6-14=-20 .
3 10 ~4 -~ 3 ~ + 2g~ 4 - 20 . R .
10
VALOR NUMERICO P 23
Va~~~ ~~~~~ic~ de ~~a e~~~e~i~~ a~geb~aica e~ e~ ~e~~~~ad~ ~~e ~e
~b~ie~e a~ ~~~~i~~i~ ~a~ ~e~~a~ ~~~ ~a~~~e~ ~~~~~ic~~ dad~~ ~ efec~~a~ de~~~~~
~a~ ~~e~aci~~e~ i~dicada~ .
f EJERCICIO 10
Red~ci~ ~~~ ~~~i~~~i~~ ~ig~ie~~e~ :
1 . 7a-9b+6a-4b .
2 . a+b-c-b-c+2c-a .
3 . 5~-11~-9+20~-1-~ .
4 . -6~+8~+5-~-~-6~-11 .
5 . -a+b+2b-2c+3a+2c-3b .
6 . -81~+19~-30~+6~+80~+~-25~ .
7 . 15a 2 -6ab-8a 2 +20-5ab- 31+a 2 _ab .
8 . -3a+4b-6a+81b-114b+31a-a-b .
9 . -71a 3 b-84a 4 b 2 +50a 3 b+84a 4 b 2 -45a 3 b+18a 3 b .
10 . -a+b-c+8+2a+2b-19-2c-3a-3-3b+3c .
11 . 1~ 2 +71~~-14~ 2 -65~~+~ 3 -~ 2 -115~ 2 +6~ 3 .
12 . ~ 4 ~-~ 3 ~2 +~ 2 ~-8~ 4 ~-~ 2 ~-10+~ 3 ~2 -7~ ;~2 -9+21~ 1 ~- ~3 +50 .
13 . 5 a ~+1-3b~+2-8 c X+3-5 a ~+1- 50+4b~+ 2 -65-b~* 2 +90+c~+ 3 +7c~+ 3 .
14 . a ~+2 -~~+ 3 -5+8-3a~+ 2 +5~~+ 3 -6+a~ , + 2- 5~~ + 3 .
15 . 0 .3a+0 .4b+0 .5c-0 .6a-0 .7 b-0 .9c+3a-3b-3c .
16 . -1 , a+ 1 b+2a-3b- ~ a- 1 b+ $ - 1 .
2 3 4 6 4 2
17 .
~~ 2 -2~~+ ~ 2 -
~
~~+2~~-2~ 2 .
10
18 . - 4 a 2 + 2 ab- e '1 2 +2-
1
.a 2 - 4 ab+ ~b 2 - 3 b 2 -2ab .
19 . 0 .4~2~+31+
~
~~2 -0 .6~ 3 - 5 ~ 2 ~-0 .2~~ 2 + ~3 -6 .
20 . 8 a~ -1 -? b~- 2+ 8 a ~-1- 1 b~-2-0 .2a~-1+ 1 b~-2 .
25 50 5 25 5

E~e~~~~~
( 1) Ha~~a~ e~ ~a~~~ ~~~~~ic~ de a 2 - 5ab + 3b 3 ~a~a a=3, b=4 .
a 2 -5ab+3b 3 =3 2 -5X3X4+3X4 3 =9-60+192=141 . R .
24 ~ ALGEBRA
30 VALOR NUMERICO DE EXPRESIONES SIMPLES
E~e~~~~~
I
(1 ) Ha~~a~ e~ ~a~~~ ~~~~~ic~ de 5ab ~a~a a = 1, b = 2 .
S~~~i~~i~~~ ~a a ~~~ ~~ ~a~~~ 1, ~ ~a b ~~~ 2, ~ ~e~d~e~~~ :
5ab=5~1 ~2=10 . R .
(2) Va~~~ ~~~~~ic~ de a 2 b 3 c4 ~a~a a=2, b=3, c = 2 .
a 2 b 3 c 4 =2 2 ~3 3 X ( ~) 4 = 4 X 27 X
1~
= 47 = 64 R .
1
(3) Va~~~ ~~~~~ic~ de 3ac ~' 2ab ~a~a a = 2, b = 9, c = 3 .
3ac/2ab=3X2~3XV2X2X9=2XV 2X6=12 . R .
4a"b 3 1 i
(4) de = b = d=3 .
Va~~~ ~~~~~ic~ 2,
3,
c=2,
5cd ~a~a
a
4 0 2
b3 4 X (J)2 X(- )3 4 X ~27
_
1/27
- =
1
R .
5cd 5 X 2 X 3 30 30 810
f EJERCICIO 11
Ha~~a~ e~ ~a~~~ ~~~~~ic~ de ~a~ e~~~e~i~~e~ ~ig~ie~~e~ ~a~a
1 1 1
a=1, b=2, c=3, ~= ~, ~= 3, ~= 4 .
1 . 3ab . 7 . ~b~e~a . 5b 2 ~2 24~~
2 . 5a 2 b 3 c .
13 . 16 .
8 . a a ~,- 1 ~c -2 ~~ 2 /~2~2
3 . b 2 ~~. 8
4 . 24~2~3~ . 9 . '/2bc 2 . Jb 3 3164b 3 ce
10 .
2
4~ ,~/ 12bc 2 .
14 .
c 2
17 .
2~
5 . a 4 b 2 ~3 . 11 . i~~ V8 a 4 ba .
3
2~
a~b 2
4a 15 . 18 .
6 . 12 .
7
c
3
~~.
12 3bc ~2 /125b~
31 VALOR NUMERICO DE EXPRESIONES COMPUESTAS

(3) Va~~~ ~~~~~ic~ de 2(2a - b) (~ 2 + ~) - (a 2 + b) (b - a) ~a~a
a=2 b=3 ~=4 ~='
La~ ~~e~aci~~e~ i~dicada~
de~~~~ de ~~~ ~a~~~~e~i~ de-
be~ efec~~a~~e a~~e~ ~~e
~i~g~~a ~~~a, a~~ :
2(2a-b)=2X(2~2-3)=2X(4-3)=2X1=2
~2+~=42 + 2
1 =16+1=161
a 2 +b=2 2 +3=4+3=7
b-a=3-2=1
Te~d~e~~~ :
2(2a-b)(~ 2 +~)-(a 2 +b)(b-a)=2X161-7X1=2X82-7=33-7=26 . R
f EJERCICIO 13
a=1, b=2, c=3, d=4, ~= 1 . ~= $, ~= 4, ~=0 .
8~ 16~
1 . (a+b)c-d . 5 . ( -1,~+8~)(a2+b2)(6~-d) .
~
9~ + b / a
.
2 . (a+b)(b-a) . 6 . (c-b)(d-c)(b-a)(~-~) .
10 . ~+~(a~+de-c ~) .
3 . (b-~)(c-~)+4a2 . 7 . b 2 (c+d)-a 2 (~+~)+2~ . 4(~+~) a 2 +b 2
4 . (2~+3~)(4~+b 2 ) 8 . 2~~~+6(b 2 +c 2 )-4d 2 . 11 . -
a c 2
VALOR NUMERICO ~ 25
2
1
(2) Va~~~ ~~~~~ic~ de
~
---+- ~a~a a=2, b=-, ~=-
4
.
3a 2 5ab b 3 ~ 2 2 5 ~ 2 ~ I~
=3--+-
---+-=
4 ~ a~
- -+
4 2X* a~
=3-20+ 1 =-16 . R .
f EJERCICIO 12
a=3, b=4, c= 3, d= Z, ~=6, ~= ~
4
1 . a 2 -2ab+b 2 . 7 .
ab ac _ bd
+
13 .
a+b - b+~
2 . c 2 +2cd+d 2 . 8 .
~ d ~
14 .
C
b-a
+
d
~-b
+ 5a .
-,/b+-~-+/6-~
.
~
12c-a
d
16~-a 1
3 .
c +
d .
9 . c ~- d 16b 2 + ~ V8d . 15 .
-
4 . 10 . 16 .
2b
V+
+ d .
~
c
-~+2 . ~~
3a - .
d~
d ~ 3 6
a 2 b 2 ~2 3c 2 4~ 2 V+ 2d /-3c + N/-8d-
5 . 3-2+ . 11 . + . 17 .
6 4 ~ 2 4
4d 2 16~ 2 2 a 2 ~2 3 ~'2+d 2
6 . 5c-1b+2d . 12 . + -1 . 18 . -a~ .
3 +
2 2
4

32 EJERCICIOS SOBRE NOTACION ALGEBRAICA
C~~ ~a~ ca~~idade~ a~geb~aica~, ~e~~e~e~~ada~ ~~~ ~e~~a~, ~~ede~ ha-
ce~~e ~a~ ~i~~a~ ~~e~aci~~e~ ~~e c~~ ~~~ ~~~e~~~ a~i~~~~ic~~ . C~~~ ~a
~e~~e~e~~aci~~ de ca~~idade~ ~~~ ~edi~ de ~~~b~~~~ ~ ~e~~a~ ~~e~e ~f~ece~
dific~~~ade~ a ~~~ a~~~~~~, ~f~ece~~~ a c~~~i~~aci~~ a~g~~~~ e~e~~~~~ .
E~e~~~~~
( 1) E~c~~ba~e ~a ~~~a de~ c~ad~ad~ de a c~~ e~ c~b~ de b .
a 2 + b 8 . R .
(2) U~ h~~b~e ~e~~a $a ; de~~~~~ ~ecibi~ $8 ~ de~~~~~ ~ag~ ~~a c~e~~a de $c .
~C~~~~~ ~e ~~eda?
Te~ie~d~ $a ~ecibi~ $8 ~~eg~ ~e~~a $(a + 8) . Si e~~~~ce~ ga~~a $c ~e ~~eda~
$(a+8- c) . R .
(3) C~~~~~ 3 ~ib~~~ a $a cada ~~~ ; 6 ~~~b~e~~~ a $b cada ~~~ ~ ~ ~~a~e~
cada ~~~ . ~C~~~~~ he ga~~ad~?
3 ~ib~~~ a $a i~~~~~a~ $3a .
6 ~~~b~e~~~ a $b i~~~~~a~ $6b .
~ ~~a~e~ a $~ i~~~~~a~ $~~ .
L~eg~ e~ ga~~~ ~~~a~ ha ~id~ de $(3a + 6b + ~~) . R .
(4) C~~~~~ ~ ~ib~~~ ig~a~e~ ~~~ $~ . ~C~~~~~ ~e ha c~~~ad~ cada ~~~?
~
Cada ~ib~~ ha c~~~ad~ $- . R .
~
(5) Te~~a $9 ~ ga~~~ $~ . ~C~~~~~ ~e ~~eda?
Me ~~eda~ $(9-~) . R .
f EJERCICIO 14
1 . E~c~~ba~e ~a ~~~a de a, b ~ ~ .
2 . E~c~~ba~e ~a ~~~a de~ c~ad~ad~ de i~, e~ c~b~ de b ~ ~a c~a~~a ~~~e~-
cia de ~ .
26 ALGEBRA
2
12 . (2~+3~+4~)(8~+6~-4~)(9~+20~) . 19 . 3(c-b) V -2(d-a) ~-
13 . c2(~+~)-d2(~+~)+b2(~+~) .
~
/6abc 3~~
20 .
cd~~
+ -
V c 2 +d 2
2
14 . . %~ . 2 ~' -
8 - b 2(b-a) abc
a ~' a 2 -~-b2
15 . (4~+2b)(18~-24~)+2(8~+2)(40~+a) . 21 . +3(a+b)(2a+3b)
b 2 -a 2
d 2 1 1 1 1 1 1
a+- 5+ 2 22 . b2+(a+b)(b+c)+(~+ ~ )2
16 ~ X
d -b ~2 23 . (2~~+3~)(4~+2c)-4~ 2 ~2 .
17 . (a+b)~/c 2 +8b-~ +8b-~V-~2 c
b2-
V c
-a+ 3 ~
18 .
( + b 1 - ( c+d ) ~. 24 . -
2 ~ab -~ b-~

NOTACION ALGEBRAICA
~ 2 7
3 . Sie~d~ a ~~ ~~~e~~ e~~e~~, e~c~~ba~~e ~~~ d~~ ~~~e~~~ e~~e~~~ c~~~e-
c~~i~~~ ~~~~e~i~~e~ a a .
4 . Sie~d~ ~ ~~ ~~~e~~ e~~e~~, e~c~~ba~~e ~~~ d~~ ~~~e~~~ c~~~ec~~i~~~
a~~e~i~~e~ a ~ .
5 . Sie~d~ ~ ~~ ~~~e~~ e~~e~~ ~a~, e~c~~ba~~e ~~~ ~~e~ ~~~e~~~ ~a~e~ c~~-
~ec~~i~~~ ~~~~e~i~~e~ a ~ .
6 . Ped~~ ~e~~a $a, c~b~~ $~ ~ ~e ~ega~a~~~ $~ . ~C~~~~~ ~ie~e Ped~~?
7 . E~c~~ba~e ~a dife~e~cia e~~~e ~ ~ ~.
8 . Deb~a ~ b~~~~a~e~ ~ ~ag~~ 6 . ~C~~~~~ deb~ ah~~a?
9 . De ~~a ~~~~ada de ~ K~ . ~a ~e ha~ ~ec~~~id~ ~ K~ . ~C~~~~~ fa~~a
~~~ a~da~?
10 . Recib~ $~ ~ de~~~~~ $a . Si ga~~~ $~, ~c~~~~~ ~e ~~eda?
11 . Te~g~ ~~e ~ec~~~e~ ~ K~ . E~ ~~~e~ a~d~ a K~ ., e~ ~a~~e~ b K~ . ~
e~ ~i~~c~~e~ c K~ . ~C~~~~~ ~e fa~~a ~~~ a~da~?
12 . A~ ~e~de~ ~~a ca~a e~ $~ ga~~ $300 . ~C~~~~~ ~e c~~~~ ~a ca~a?
13 . Si ha~ ~~a~~c~~~id~ ~ d~a~ de ~~ a~~, ~c~~~~~~ d~a~ fa~~a~ ~~~ ~~a~~c~~~i~?
14 . Si ~~ ~~~b~e~~ c~e~~a $a, Ic~~~~~ i~~~~~a~~~ 8 ~~~b~e~~~ ; 15 ~~~b~e-
~~~; ~ ~~~b~e~~~?
15 . E~c~~ba~e ~a ~~~a de~ d~~~~ de a c~~ e~ ~~i~~~ de b ~ ~a ~i~ad de c .
16 . E~~~e~a~ ~a ~~~e~ficie de ~~a ~a~a ~ec~a~g~~a~ ~~e ~ide a ~ . de ~a~g~
~ b ~. de a~ch~ .
17 . U~a e~~e~~i~~ ~ec~a~g~~a~ de 23 ~ . de ~a~g~ ~ide ~ ~ . de a~ch~ . E~-
~~e~a~ ~~ ~~~e~ficie .
18 . ~C~~~ ~e~~ ~a ~~~e~ficie de ~~ c~ad~ad~ de ~ ~ . de ~ad~?
19 . Si ~~ ~~~b~e~~ c~e~~a $a ~ ~~ ~~a~e $b, ~c~~~~~ i~~~~~a~~~ 3 ~~~b~e~~~
~ 6 ~~a~e~?, ~~ ~~~b~e~~~ ~ ~ ~~a~e~?
20 . E~c~~ba~e e~ ~~~d~c~~ de a + b ~~~ ~ + ~ .
21 . Ve~d~ (~ + 6) ~~a~e~ a $8 cada ~~~ . ~C~~~~~ i~~~~~a ~a ~e~~a?
22 . C~~~~~ (a - 8) caba~~~~ a (~ + 4) b~~~~a~e~ cada ~~~ . ~C~~~~~ i~~~~~a
~a c~~~~a?
23 . Si ~ ~~~ice~ c~e~~a~ 75 ~~c~e~ ; ~c~~~~~ c~e~~a ~~ ~~~i~?
24 . Si ~~~ $a c~~~~~ ~ ki~~~ de a~~ca~, ~c~~~~~ i~~~~~a ~~ ki~~?
25 . Se c~~~~a~ (~ - 1) caba~~~~ ~~~ 3000 c~~~~e~ . ~C~~~~~ i~~~~~a cada
caba~~~?
26 C~~~~~ a ~~~b~e~~~ ~~~ ~ ~~~e~ . ~A c~~~ hab~~a ~a~id~ cada ~~~b~e~~
~i h~bie~a c~~~~ad~ 3 ~e~~~ ~~~ e~ ~i~~~ ~~eci~?
27 . La ~~~e~ficie de ~~ ca~~~ ~ec~a~g~~a~ e~ ~ ~ . 2 ~ e~ ~a~g~ ~ide 14 ~ .
E~~~e~a~ e~ a~ch~ .
28 . Si ~~ ~~e~ ha ~ec~~~id~ ~ + 1 K~ . e~ a h~~a~, ~c~~~ e~ ~~ ~e~~cidad ~~~
h~~a?
29 . Te~~a $a ~ c~b~~ $b . Si e~ di~e~~ ~~e ~e~g~ ~~ e~~~e~ ~~d~ e~ c~~~~a~
(~ - 2) ~ib~~~, ~a c~~~ ~a~e cada ~ib~~?
30 E~ e~ ~i~~ ba~~ de ~~ h~~e~ ha~ ~ habi~aci~~e~ . E~ e~ ~eg~~d~ ~i~~ ha~
d~b~e ~~~e~~ de habi~aci~~e~ ~~e e~ e~ ~~i~e~~ ; e~ e~ ~e~ce~~ ~a ~i~ad
de ~a~ ~~e ha~ e~ e~ ~~i~e~~ . ~C~~~~a~ habi~aci~~e~ ~ie~e e~ h~~e~?
31 . Ped~~ ~ie~e a ~~c~e~ ; J~a~ ~ie~e ~a ~e~ce~a ~a~~e de ~~ de Ped~~ ; E~~i~~e
~a c~a~~a ~a~~e de~ d~~~~ de ~~ de Ped~~ . La ~~~a de ~~ ~~e ~ie~e~
~~~ ~~e~ e~ ~e~~~ ~~e 1000 ~~c~e~ . ~C~~~~~ fa~~a a e~~a ~~~a ~a~a ~e~
ig~a~ a 1000 ~~c~e~?

2 8 ~ ALGEBRA
NOTAS SOBRE EL CONCEPTO DE NUMERO
E~ c~~ce~~~ de ~~~e~~ ~a~~~a~ (~~a~e A~i~~~~ica Te~~ic~-P~~c~ica, 33),
~~e ~a~i~face ~a~ e~ige~cia~ de ~a A~i~~~~ica e~e~e~~a~ ~~ ~e~~~~de a ~a ge~e-
~a~i~aci~~ ~ ab~~~acci~~ ca~ac~e~~~~ica~ de ~a ~~e~a~~~ia a~geb~aica .
E~ A~geb~a ~e de~a~~~~~a ~~ c~~c~~~ de ~a~ide~ ge~e~a~ a~~icab~e a c~a~-
~~ie~ ~i~~ e~~ecia~ de ~~~e~~ . C~~~ie~e ~~e~, c~~~ide~a~ c~~~ ~e ha a~~~iad~
e~ ca~~~ de ~~~ ~~~e~~~ ~~~ ~a i~~~~d~cci~~ de ~~e~~~ e~~e~, ~~e ~a~i~face~
~a~ ~e~e~ ~~e ~eg~~a~ ~a~ ~~e~aci~~e~ f~~da~e~~a~e~, ~a ~~e, c~~~ ~e~e~~~
~~~ ade~a~~e, e~ ~~~e~~ ~a~~~a~ (1) ~~ ~~~ ~i~~e ~a~a efec~~a~ ~a ~e~~a ~ ~a
di~i~i~~ e~ ~~d~~ ~~~ ca~~~ . Ba~~e ~~~ e~ ~~~e~~~, dad~ e~ ~i~e~ ~a~e~~~ic~
~~e a~ca~~a~e~~~ a ~~ ~a~g~ de e~~e ~e~~~, e~~~ica~ c~~~ ~e ha ~~egad~ a~
c~~ce~~~ de ~~~e~~ ~ea~ .
Pa~a hace~ ~~~ c~~~~e~~ib~e ~a a~~~iaci~~ de~ ca~~~ de ~~~ ~~~e~~~,
ad~~~a~e~~~ ~~ d~b~e c~i~e~i~ . P~~ ~~ ~ad~, ~~ c~i~e~i~ hi~~~~ic~ ~~e ~~~ haga
c~~~ce~ ~a g~ad~a~ a~a~ici~~ de ~a~ di~~i~~a~ c~a~e~ de ~~~e~~~ ; ~~~ ~~~~, ~~
c~i~e~i~ i~~~i~i~~ ~~e ~~~ ~~~ga de ~a~ifie~~~ c~~~ cie~~a~ ~ece~idade~ ~a~e-
~ia~e~ ha~ ~b~igad~ a ~~~ ~a~e~~~ic~~ a i~~~~d~ci~ ~~e~~~ e~~e~ ~~~~~ic~~ .
E~~e d~b~e c~i~e~i~, ~~~~ificab~e ~~~ ~a ~~d~~e did~c~ica de e~~e ~ib~~, ~e~~i~i~~
a~ ~~i~ci~ia~~e a~ca~~a~ ~~a c~~~~e~~i~~ c~a~a de~ c~~ce~~~ f~~~a~ (ab~~~ac~~)
de ~~~ ~~~e~~~ ~ea~e~ .
EL NUMERO ENTERO Y EL NUMERO FRACCIONARIO
M~ch~ a~~e~ de ~~e ~~~ g~ieg~~ (E~d~~i~, E~c~ide~, A~~~~~i~, e~c .) ~ea-
~i~a~a~ ~a ~i~~e~a~i~aci~~ de ~~~ c~~~ci~ie~~~~ ~a~e~~~ic~~, ~~~ babi~~~i~~
(~eg~~ ~~e~~~a~ ~a~ ~ab~i~~a~ c~~eif~~~e~ ~~e da~a~ de 2000-1800 A .C .) ~ ~~~
egi~ci~~ (c~~~ ~e ~e e~ e~ ~a~i~~ de Rhi~d) c~~~c~a~ ~a~ f~acci~~e~ .
La ~ece~idad de ~edi~ ~ag~i~~de~ c~~~i~~a~ ~a~e~ c~~~ ~a ~~~gi~~d, e~
~~~~~e~, e~ ~e~~, e~c ., ~~e~~ a~ h~~b~e a i~~~~d~ci~ ~~~ ~~~e~~~ f~acci~~a~i~~ .
C~a~d~ ~~~a~~~ ~~a ~~idad c~a~~~ie~a, ~~~ e~e~~~~, ~a ~a~a, ~a~a
~edi~ ~~a ~ag~i~~d c~~~i~~a (~ag~i~~d e~ca~a~ ~ ~i~ea~), ~~ede ~c~~~i~ ~~a
de e~~a~ d~~ c~~a~ : ~~e ~a ~~idad e~~~ c~~~e~ida ~~ ~~~e~~ e~~e~~ de ~ece~,
~ ~~e ~~ e~~~ c~~~e~ida ~~ ~~~e~~ e~~e~~ de ~ece~ .(' .,) E~ e~ ~~i~e~ ca~~,
~e~~e~e~~a~~~ e~ ~e~~~~ad~ de ~a ~edici~~ c~~ ~~ ~~~e~~ e~~e~~ . E~ e~ ~e-
g~~d~ ca~~, ~e~d~e~~~ ~~e f~acci~~a~ ~a ~~idad e~egida e~ d~~, e~ ~~e~, ~ e~
c~a~~~ ~a~~e~ ig~a~e~ ; de e~~e ~~d~, ha~~a~e~~~ ~~a f~acci~~ de ~a ~~idad
~~e e~~~ c~~~e~ida e~ ~a ~ag~i~~d ~~e ~~a~a~~~ de ~edi~ . E~ ~e~~~~ad~ de e~~a
~~~i~a ~edici~~ ~~ e~~~e~a~~~ c~~ ~~ ~a~ de ~~~e~~~ e~~e~~~, di~~i~~~~ de
ce~~, ~~a~ad~~ ~e~~ec~i~a~e~~e ~~~e~ad~~ ~ de~~~i~ad~~ . E~ de~~~i~ad~~
~~~ da~~ e~ ~~~e~~ de ~a~~e~ e~ ~~e he~~~ di~idid~ ~a ~~idad, ~ e~ ~~~e-
~ad~~, e~ ~~~e~~ de ~~b~~idade~ c~~~e~ida~ e~ ~a ~ag~i~~d ~~e acaba~~~
de ~edi~ . S~~ge~ de e~~e ~~d~ ~~~ ~~~e~~~ f~acci~~a~i~~ . S~~ ~~~e~~~ f~ac-
ci~~a~i~~ 1/2 . 1/3 . 3/5, e~c .
(1) P . L . G . Di~ich~e~ (a~e~~~, 1805-1859), ha ~~~~e~id~ ~~e ~~ e~ ~ece~a~ia~e~~e i~di~-
~e~~ab~e a~~~ia~ e~ c~~ce~~~ de ~~~e~~ ~a~~~a~, ~a ~~e -~eg~~ ~~- c~a~~~ie~ ~~i~ci~i~
de ~a ~~~ a~~a ~a~e~~~ica ~~ede de~~~~~a~~e ~~~ ~edi~ de ~~~ ~~~e~~~ ~a~~~a~e~ .
(2) E~ ~a ~~~c~ica ~ hab~a~d~ c~~ ~ig~~, ~i~g~~a ~edida ~e~~~~a e~ac~a, e~ ~a~~~ de
~~ i~~e~fec~~ de ~~e~~~~~ i~~~~~~e~~~~ de ~edida ~ de ~~e~~~~~ ~e~~id~~ .

P~de~~~ deci~ ~a~bi~~, ~~e ~~~ ~~~e~~~ f~acci~~a~i~~ ~~~ ~~e ~~~ ~e~-
~i~e~ e~~~e~a~ e~ c~cie~~e de ~~ia di~i~i~~ i~e~ac~a, ~ ~~ ~~e e~ ~~ '~~i~~~, ~~a
di~i~i~~ e~ ~a c~a~ e~ di~ide~d~ ~~ e~ ~~~~i~~~ de~ di~i~~~ .
C~~~ ~e ~e, e~ ~~~~ici~~ a ~~~ ~~~e~~~ f~acci~~a~i~~ ~e~e~~~~ ~~~ ~~-
~e~~~ e~~e~~~, ~~e ~~de~~~ defi~i~ c~~~ a~~e~~~~ ~~e e~~~e~a~ e~ c~cie~~e
de ~~a di~i~i~~ e~ac~a, c~~~ ~~~ e~e~~~~, 1, 2, 3, e~c .
5L5 Si 4 6 : 2-- :1 .
0 1 0 2
EL NUMERO RACIONAL Y EL NUMERO IRRACIONAL
Sig~ie~d~ e~ ~~de~ hi~~~~ic~ ~~e ~~~ he~~~ ~~a~ad~, ~a~a~~ a ~e~ ah~~a
c~~~d~ ~ c~~~ ~~~gie~~~ ~~~ ~~~e~~~ i~~aci~~a~e~ .
E~ i~d~dab~e ~~e f~e~~~ ~~~ g~ieg~~ ~~ie~e~ c~~~cie~~~ ~~i~e~~ ~~~ ~~-
~e~~~ i~~aci~~a~e~ . L~~ hi~~~~iad~~e~ de ~a ~a~e~~~ica, e~~~~ de ac~e~d~ e~
a~~ib~i~ a Pi~~g~~a~ de Sa~~~ (540 A .C.), e~ de~c~b~i~ie~~~ de e~~~~ ~~~e~~~,
a~ e~~ab~ece~ ~a ~e~aci~~ e~~~e e~ ~ad~ de ~~ c~ad~ad~ ~ ~a diag~~a~ de~ ~i~~~ .
M~~ ~a~de, Te~d~~~ de Ci~e~e (400 A .C .), ~a~e~~~ic~ de ~a e~c~e~a ~i~ag~-
~ica, de~~~~~~ ge~~~~~ica~e~~e ~~e --,/ _2, ~ _3, 'Y/75, V7, e~c ., ~~~ i~~aci~~a~e~ .
E~c~ide~ (300 A .C .), e~~~di~ e~ e~ Lib~~ X de ~~~ "E~e~e~~~~", cie~~a~
~ag~i~~de~ ~~e a~ ~e~ ~edida~ ~~ e~c~~~~a~~~ ~i~g~~ ~~~e~~ e~~e~~ ~i
f~acci~~a~i~ ~~e ~a~ e~~~e~e . E~~a~ ~ag~i~~de~ ~e ~~a~a~ i~c~~~e~~~~ab~e~, ~
~~~ ~~~e~~~ ~~e ~e ~~igi~a~ a~ ~edi~ ~a~e~ ~ag~i~~de~ ~e ~~a~a~ i~~aci~~a~e~ . ( >
E~e~~~~~ de ~a~e~ ~ag~i~~de~ ~~~ ~a ~e~aci~~ de~ ~ad~ (~e ~~ c~ad~ad~ c~~
~a diag~~a~ de~ ~i~~~, ~~e ~e e~~~e~a c~~ e~ ~~~e~~ i~~aci~~a~ ~/~ 2 + b' '
2 ;
~ ~a ~e~aci~~ de ~a ci~c~~fe~e~cia, a~ di~~e~~~ ~~e ~e e~~~e~a c~~ ~a ~e~~a
7c = 3 .141592 . . .
a
d =~ a ' + D ~
C
C =I~ =3 .14159
~ 29
(, ;) A~ e~~~~e~ ~i~~e~~~ica~e~~e ~~~ ~~~e~~~ i~~aci~~a~e~, E~c~ide~ ~~~ ~~a~~ a~~~~e~~~~,
~ a ~~~ ~aci~~a~e~ ~~~ ~~a~~ ~~~~e~~~~, ~a~ab~a~ ~~e ~ig~ifica~ ~i~ ~edida ~ c~~ ~edida .
Pa~a ~e~a~a~ e~ hech~ de ~~e e~~~~ ~~~e~~~ (~~~ i~~aci~~a~e~) ~~ ~e~~a~ e~~~e~i~~ ~~~ de~ig~aba
c~~ ~a ~~~ a~~g~~. B~eci~ (475-554 D . C .), a~ ~~ad~ci~ e~~~e~ c~~i~e~~~~abi~i~ e i~c~~~e~-
~~~abi~i~ . Si~ e~ba~g~, Ge~a~d~ de C~e~~~a (1114-1187), e~ ~~a ~~ad~cci~~ (~e ~~ c~~e~~a~i~
~~abe ~~b~e E~c~ide~, ~~i~i~~ e~~~~ea~e~~e ~a~i~~a~i~ e i~~a~i~~a~i~, a~ ~~~a~ ~~g~~ ~ a~~g~~
c~~~ ~a~~~ ~ ~~ e~ ~a ace~ci~~ de ~a~ab~a (~e~b~~), ~~ada ~~~ E~c~ide~ . E~~e e~~~~ ~e
dif~~di~ a ~~ ~a~g~ de ~~da ~a Edad Media, ~~e~a~ecie~d~ e~ ~~e~~~~~ d~a~ e~ ~~~b~e de
~~~e~~~ i~~aci~~a~e~ .
FIGURA 1
C = ci~c~~fe~e~cia
D =di~~e~~~

3 0 ALGEBRA
C~~~ c~~~ec~e~cia de ~a i~~~~d~cci~~ de ~~~ ~~~e~~~ i~~aci~~a~e~, c~~-
~ide~a~~~ ~aci~~a~e~ e~ c~~~~~~~ de ~~~ ~~~e~~~ f~acci~~a~i~~ ~ e~ c~~~~~~~
de ~~~ ~~~e~~~ e~~e~~~ . Defi~i~~~ e~ ~~~e~~ ~aci~~a~ c~~~ a~~e~ ~~~e~~
~~e ~~ede e~~~e~a~~e c~~~ c~cie~~e de d~~ e~~e~~~ . Y e~ ~~~e~~ i~~aci~~a~ c~~~
a~~e~ ~~~e~~ ~ea~ ~~e ~~ ~~ede e~~~e~a~~e c~~~ e~ c~cie~~e de d~~ e~~e~~~ .
L~a~a~~~ ~~~e~~ ~ea~e~ a~ c~~~~~~~ de ~~~ ~~~e~~~ ~aci~~a~e~ e i~~a-
ci~~a~e~ .
LOS NUMEROS POSITIVOS Y NEGATIVOS
L~~ ~~~e~~~ ~ega~i~~~ ~~ f~e~~~ c~~~cid~~ ~~~ ~~~ ~a~e~~~ic~~ de ~a
a~~ig~edad, ~a~~~ e~ e~ ca~~ de Di~fa~~~ (~ig~~ III D . C .?), ~~e e~ ~~ A~i~~~~ica,
a~ e~~~ica~ e~ ~~~d~c~~ de d~~ dife~e~cia~, i~~~~d~ce ~~ ~~~e~~ c~~ ~ig~~ + .
E~ e~ ~ig~~ VI, ~~~ hi~d~e~ B~ah~ag~~~a ~ Bh~~ka~a ~~a~ ~~~ ~~~e~~~ ~ega~i~~~
de ~~ ~~d~ ~~~c~ic~, ~i~ ~~ega~ a da~ ~~a defi~ici~~ de e~~~~ . D~~a~~e ~a
Edad Media ~ e~ Re~aci~ie~~~ ~~~ ~a~e~~~ic~~ ~eh~~e~~~ ~~a~ ~~~ ~~~e~~~
~ega~i~~~, ~ f~e Ne~~~~ e~ ~~i~e~~ e~ c~~~~e~de~ ~a ~e~dade~a ~a~~~a~e~a de
e~~~~ ~~~e~~~ . P~~~e~i~~~e~~e Ha~~i~~ (1560-1621) i~~~~d~~~ ~~~ ~ig~~~ + ~ -
~a~a ca~ac~e~i~a~ ~~~ ~~~e~~~ ~~~i~i~~~ ~ ~ega~i~~~ .
La ~ig~ificaci~~ de ~~~ ~~~e~~~ ~e~a~i~~~ ~ c~~ ~ig~~~ (~~~i~i~~~ ~ ~ega-
~i~~~) ~e c~~~~e~de c~a~a~e~~e, c~a~d~ ~~~ ~~i~i~a~~~ ~a~a ~e~~e~e~~a~ e~
~e~~~~ad~ de ~edi~ ~ag~i~~de~ ~e~a~i~a~, e~ deci~, ~ag~i~~de~ c~~a~ ca~~idade~
~~ede~ ~~~a~~e e~ ~e~~id~~ ~~~e~~~~, ~a~ c~~~ ~~cede c~a~d~ ~~a~a~~~ de
~edi~ ~a ~~~gi~~d ge~g~~fica de ~~a ~egi~~ de~e~~i~ada ; ~ de e~~~e~a~ e~
g~ad~ de ~e~~e~a~~~a de ~~ ~~ga~ dad~ . E~ e~ ~~i~e~ ca~~, ~~de~~~ hab~a~
de ~~~gi~~d e~~e ~ ~e~~e c~~ ~e~~ec~~ a ~~ ~e~idia~~ fi~ad~ a~bi~~a~ia~e~~e
(G~ee~~ich) . E~ e~ ~eg~~d~ ca~~, ~~de~~~ ~efe~i~~~~ a g~ad~~ ~~b~e ce~~ ~
g~ad~~ ba~~ ce~~ . C~~~e~ci~~a~~e~~e fi~a~~~ ~~~ ~~~e~~~ ~~~i~i~~~ ~ c~~
~ig~~ + e~ ~~a di~ecci~~, ~ ~~~ ~~~e~~~ ~ega~i~~~ ~ c~~ ~ig~~ -, e~ ~a di~ec-
ci~~ ~~~e~~a .
Si ~~b~e ~~a ~e~i~~ec~a fi~a~~~ ~~ ~~~~~ ce~~, a ~a~~i~ de~ c~a~, hacia ~a
de~echa, ~e~a~a~~~ ~~~~~~ ~~e ~e~~e~e~~a~ ~~a de~e~~i~ada ~~idad, ~~~ ~e-
~~~~a~ ~~~ ~~~~~~ A, B, C, e~c . Si ~~b~e e~a ~i~~a ~e~i~~ec~a, a ~a~~i~ de~ ~~~~~
ce~~ (~~a~ad~ ~~ige~), ~~~cede~~~ de~ ~i~~~ ~~d~ hacia ~a i~~~ie~da, ~e~d~e-
~~~ ~~~ ~~~~~~ a, b, c, e~c . Si c~~~e~i~~~ e~ ~~e ~~~ ~~~~~~ de ~a ~e~i~~ec~a i~di-
cad~~ a ~a de~echa de~ ~~~~~ ce~~ ~e~~e~e~~a~ ~~~e~~~ ~~~i~i~~~ (A, B, C, e~c .) ;
~~~ ~~~~~~ ~e~a~ad~~ a ~a i~~~ie~da (a, b, c, e~c .), ~e~~e~e~~a~~~ ~~~e~~~
~ega~i~~~ .
c b a
I
A B C
-3 -2 -1 0 +1 +2 +3
Hi~~~~ica~e~~e, ~~~ ~~~e~~~ ~ega~i~~~ ~~~ge~ ~a~a hace~ ~~-
~ib~e ~a ~e~~a e~ ~~d~~ ~~~ ca~~~ . De e~~e ~~d~, ~a ~e~~a ~e c~~~ie~~e e~ ~~a
~~e~aci~~ i~~e~~a de ~a ~~~a, ~ ~e hace ~~~ib~e ~e~~a~~e a ~~ ~i~~e~d~ ~e~~~
~~ ~~~~~ae~d~ ~a~~~ .

L~~ ~~~e~~~ ~ ~~~ ~~~b~~~~ ~i~e~a~e~ ~ega~i~~~ ~e di~~i~g~e~ ~~~ e~ ~ig~~ -
~~e ~~e~a~ a~~e~~e~~~ . L~~ ~~~e~~~ ~~~i~i~~~ ~ ~~ ~e~~e~e~~aci~~ ~i~e~a~ ~~e~a~
e~ ~ig~~ +, ~ie~~~e ~~e ~~ i~icie~ ~~a e~~~e~i~~ a~geb~aica .
E~ ~~~e~~ ce~~ . C~a~d~ ~~a~a~~~ de a~~ehe~de~ e~ c~~ce~~~ de ~~~e~~
~a~~~a~, ~e~~~ c~~~ ~~~e ~~~ge de ~a c~~~a~aci~~ de c~~~~~~~~ e~~i~a~e~~e~
~ c~~~di~ab~e~ e~~~e ~~ . P~~ e~~e~~i~~ ~~a~a~~~ c~~~~~~~ a~ ~~e ~ie~e ~~ ~~~~
e~e~e~~~ ~ ~~e ~e ~e~~e~e~~a ~~~ e~ ~~~e~~ 1 . Ah~~a, c~~~ide~a~~~ e~ ~~~e~~
ce~~ c~~~ e~~~e~i~~ de ~~ c~~~~~~~ ~~~~ ~ ~ac~~, e~ deci~, ~~ c~~~~~~~ ~~e
ca~ece de e~e~e~~~~ .
P~~ ~~~a ~a~~e, e~ ce~~ ~e~~e~e~~a ~~ e~e~e~~~ de ~e~a~aci~~ e~~~e ~~~
~~~e~~~ ~ega~i~~~ ~ ~~~i~i~~~, de ~~d~ ~~e e~ ce~~ e~ ~a~~~ ~~e c~a~~~ie~
~~~e~~ ~ega~i~~ ~ ~e~~~ ~~e c~a~~~ie~ ~~~e~~ ~~~i~i~~ .
E~ ~ig~ie~~e diag~a~a ~~~ ac~a~a~~ ~a~ di~~i~~a~ c~a~e~ de ~~~e~~~ c~~
~~~ c~a~e~ ~a~~~ a ~~aba~a~ :
NUMEROS REALES
I
~
0
Nega
I
~i~~~ Ce~~ P~~i~i~~~
1 1 1
I
Raci~~a~e~ I~~aci~~a~e~ Raci~~a~e~ I~~aci~~a~e~
E~~e~~~ F~acci~~a~i~~ E~~e~~~' ~ ~a~~~a~i~~
LEYES FORMALES DE LAS OPERACIONES FUNDAMENTALES
CON NUMEROS REALES
He~~~ ~i~~~ ~~~a~ia~e~~e c~~~ a ~~a~~~ de~ c~~~~ de ~a hi~~~~ia de ~a~
~a~e~~~ica~, ~e ha id~ a~~~ia~d~ ~~ce~i~a~e~~e e~ ca~~~ de ~~~ ~~~e~~~,
ha~~a ~~ega~ a~ c~~ce~~~ de ~~~e~~ ~ea~. E~ ca~i~~ ~ec~~~id~ ha ~id~, ~~a~
~ece~, e~ ge~~~~~ic~, ~~e ~ie~~~e de~e~b~ca e~ ~a A~i~~~~ica ~~~a, f~~~a~ ;
~~~a~ ~ece~, e~ ca~i~~ ~~~~, f~~~a~ ha i~iciad~ e~ ~ec~~~id~ ~a~a de~e~b~ca~
e~ ~~ i~~~i~i~~, e~ ~~ ge~~~~~ic~ . C~~~ e~e~~~~~ de~ ~~i~e~ ca~~, ~e~e~~~
~~~ ~~~e~~~ i~~aci~~a~e~, i~~~~d~cid~~ c~~~ ~a~~~ de d~~ ~eg~e~~~~ c~~ e~
~~~~~~i~~ de ~e~~e~e~~a~ ~ag~i~~de~ i~c~~~e~~~~ab~e~, ~ ~~e hace~ ~~~ib~e
~a e~~~e~i~~ de~ ~e~~~~ad~ de ~a ~adicaci~~ i~e~ac~a . Y ~a~bi~~, ~~~ ~~~e~~~
f~acci~~a~i~~ ~~e ~~~ge~ ~a~a e~~~e~a~ e~ ~e~~~~ad~ de ~edi~ ~ag~i~~de~ c~~-
~e~~~~ab~e~, ~ ~~e hace~ ~~~ib~e ~a di~i~i~~ i~e~ac~a, C~~~ e~e~~~~ de~
~eg~~d~ ca~~, e~~~~ ~~~ ~~~e~~~ ~ega~i~~~ ~~e a~a~ece~ ~~~ ~~i~e~a ~e~ c~~~
~a~ce~ de ec~aci~~e~, ~ hace~ ~~~ib~e ~a ~e~~a e~ ~~d~~ ~~~ ca~~~, ~a ~~e c~a~d~
e~ ~i~~e~d~ e~ ~e~~~ ~~e e~ ~~~~~ae~d~ e~~a ~~e~aci~~ ca~ece de ~e~~id~
c~a~d~ ~~aba~a~~~ c~~ ~~~e~~~ ~a~~~a~e~ . M~~ ~a~de, e~~~~ ~~~e~~~ ~ega~i~~~
(~e~a~i~~~) ~e~~i~~~ ~a~a e~~~e~a~ ~~~ ~~~~~~ a ~~~ ~ ~~~~ ~ad~ de ~~a ~ec~a
i~defi~ida .
Si~ ~~e~e~~i~~e~ de ~~~f~~di~a~ ~~e~a~~~a~e~~e e~ e~ ca~~~ ~~~~~ic~,
~a~~~ a e~~~~e~ ~a~ ~e~e~ f~~~a~e~ (e~~~ e~, ~~e ~~ ~~~a~ e~ c~e~~a ~a ~a~~-
~a~e~a de ~~~ ~~~e~~~) de ~a ~~~a ~ de ~a ~~~~i~~icaci~~, ~a ~~e ~a~ de~~~ ~~e-
~aci~~e~ f~~da~e~~a~e~ ~~ede~ e~~~ica~~e c~~~ i~~e~~a~ de ~~~a~, a~~, ~a ~e~~a,
~ 31

32 40
~a di~i~i~~, ~a ~~~e~ciaci~~, ~a ~~ga~i~~aci~~ ~ ~a ~adicaci~~ . C~~~ie~e i~
ada~~a~d~ ~a ~e~~a~idad de~ ~~i~ci~ia~~e a~ ca~~c~e~ f~~~a~ (ab~~~ac~~) de e~~a~
~e~e~, ~~e~ e~~~ c~~~~ib~i~~ a ~a c~~~~e~~i~~ de ~~~ ~~~b~e~a~ ~~e ~~~e~i~~~e~~e
~e ~~a~~ea~~~ ~a~ ~a~e~~~ica~ ~~~e~i~~e~ . P~~ ~~~a ~a~~e, e~ c~~~~~~~ de e~~a~
~e~e~ f~~~a~e~ c~~~~i~~i~~ ~~a defi~ici~~ i~di~ec~a de ~~~ ~~~e~~~ ~ea~e~ ~ de
~a~ ~~e~aci~~e~ f~~da~e~~a~e~ . E~~a~ ~e~e~ ~~e ~~ ~e~~ie~e~ de~~~~~aci~~, ~~e~
~~~ de a~~ehe~~i~~ i~~edia~a, ~e ~~a~a~ a~i~~a~ .
IGUALDAD
I . A~i~~a de ide~~idad : a = a .
II . A~i~~a de ~eci~~~cidad : ~i a = b, ~e~e~~~ ~~e b = a .
III . A~i~~a de ~~a~~i~i~idad : ~i a = b ~ b = c, ~e~e~~~ ~~e a = c .
SUMA O ADICION
1 . A~i~~a de ~~if~~~idad : ~a ~~~a de d~~ ~~~e~~~ e~ ~ie~~~e ig~a~,
e~ deci~, ~~ica ; a~~, ~i a = b ~ c = d, ~e~e~~~ ~~e a + c = b + d .
II . A~i~~a de c~~~~~a~i~idad : a + b = b + a .
III . A~i~~a de a~~cia~i~idad : (a + b) + c = a + (b + c) .
IV . A~i~~a de ide~~idad, ~ ~~d~~~ de ~a ~~~a: ha~ ~~ ~~~e~~ ~ ~~~~
~~ ~~~e~~, e~ ce~~, de ~~d~ ~~e a + 0 = 0 + a = a, ~a~a c~a~~~ie~ ~a~~~ de a .
De ah~ ~~e e~ ce~~ ~eciba e~ ~~~b~e'de e~e~e~~~ id~~~ic~ ~ ~~d~~~ de ~a ~~~a .
ALGEBRA
MULTIPLICACION
I . A~i~~a de ~~if~~~idad : e~ ~~~d~c~~ de d~~ ~~~e~~~ e~ ~ie~~~e ig~a~,
e~ deci~, ~~ic~, a~~ ~i a = b ~ c = d, ~e~e~~~ ~~e ac = bd .
II . A~i~~a de c~~~~~a~i~idad : ab = ba .
III . A~i~~a de a~~cia~i~idad : (ab) c = a (bc) .
IV . A~i~~a de di~~~ib~~i~idad : c~~ ~e~~ec~~ a ~a ~~~a ~e~e~~~ ~~e
a (b + c) = ab + ac .
V . A~i~~a de ide~~idad, ~ ~~d~~~ de~ ~~~d~c~~ : ha~ ~~ ~~~e~~ ~ ~~~~
~~ ~~~e~~, e~ ~~~ (1), de ~~d~ ~~e a .1 = 1 . a = a, ~a~a c~a~~~ie~ ~a~~~ de a .
VI . A~i~~a de e~i~~e~cia de~ i~~e~~~ : ~a~a ~~d~ ~~~e~~ ~ea~ a 7~= 0
(a di~~i~~~ de ce~~) c~~~e~~~~de ~~ ~~~e~~ ~ea~, ~ ~~~~ ~~~, ~, de ~~d~ ~~e
a~ = 1 . E~~e ~~~e~~ ~ ~e ~~a~a i~~e~~~ ~ ~ec~~~~c~ de a, ~ ~e ~e~~e~e~~a ~~~ 1/a .
AXIOMAS DE ORDEN
I . T~ic~~~~~a : Si ~e~e~~~ d~~ ~~~e~~~ ~ea~e~ a ~ b ~~~~ ~~ede habe~ ~~a
~e~aci~~, ~ ~~~~ ~~a, e~~~e a~b~~, ~~e a > b ; a = b ~ a < b .
M~~~~~~~a de ~a ~~~a : ~i a > b ~e~e~~~ ~~e a + c > b + c .
M~~~~~~~a de ~a ~~~~i~~icaci~~ : ~i a > b ~ c > 0 ~e~e~~~ ~~e ac > bc .

AXIOMA DE CONTINUIDAD
1 . Si ~e~e~~~ d~~ c~~~~~~~~ de ~~~e~~~ ~ea~e~ A ~ B, de ~~d~ ~~e ~~d~
~~~e~~ de A e~ ~e~~~ ~~e c~a~~~ie~ ~~~e~~ de B, e~i~~i~~ ~ie~~~e ~~ ~~~e~~
~ea~ c c~~ e~ ~~e ~e ~e~ifi~~e a :5 c :5 b, e~ ~~e a e~ ~~ ~~~e~~ ~~e e~~~
de~~~~ de~ c~~~~~~~ A, ~ b e~ ~~ ~~~e~~ ~~e e~~~ de~~~~ de~ c~~~~~~~ B .
~J!'!-1 ~: : ~. !C . ~, ENTLES CON LOS NUMEROS RELATIVOS
SUMA DE NUMEROS RELATIVOS
E~ ~a ~~~a ~ adici~~ de ~~~e~~~ ~e~a~i~~~ ~~de~~~ c~~~ide~a~ c~a~~~
ca~~~ : ~~~a~ d~~ ~~~e~~~ ~~~i~i~~~ ; ~~~a~ d~~ ~~~e~~~ ~ega~i~~~ ; ~~~a~ ~~
~~~i~i~~ c~~ ~~~~ ~ega~i~~, ~ ~~~a~ e~ ce~~ c~~ ~~ ~~~e~~ ~~~i~i~~ ~ ~ega~i~~ .
I) de (~i~ ~~~ii i~, ~~ i i~~,
Reg~a
Pa~a ~~~a~ d~~ ~~~e~~~ ~~~i~i~~~ ~e ~~~cede a ~a ~~~a (+4)+(+2)=+6
a~i~~~~ica de ~~~ ~a~~~e~ ab~~~~~~~ de a~b~~ ~~~e~~~, ~ a~
~e~~~~ad~ ~b~e~id~ ~e ~e a~~e~~~e e~ ~ig~~ + . A~~ ~e~e~~~ :
P~de~~~ ~e~~e~e~~a~ ~a ~~~a de d~~ ~~~e~~~ ~~~i~i~~~ de~ ~ig~ie~~e ~~d~ :
-4 3
'') S~~a de d~~ ~~~e~~~ ~ega~i~~~
Reg~a
Pa~a ~~~a~ d~~ ~~~e~~~ ~ega~i~~~ ~e ~~~cede a ~a ~~~a (- 4) + (- 2) _ - 6
a~i~~~~ica de ~~~ ~a~~~e~ ab~~~~~~~ de a~b~~, ~ a~ ~e~~~~ad~
~b~e~id~ ~e ~e a~~e~~~e e~ ~ig~~ - . A~~ ~c~~ic~~~ :_ __
P~de~~~ ~e~~e~e~~a~ ~a ~~~a de d~~ ~~~e~~~ ~ega~i~~~ de~ ~ig~ie~~e
~~~~~:
~~~~~~~ ~~~~~ . ~
-1 0 +~
+4-
+Y
FIGURA 2
E- - 2 4
- 7 - 6 - S 4 -3 - 1 0 +1 2 1 3 +4
FIGURA 3
+6 ---- T
+3
A
i
+4
+ 2 -~
+5 i-6 +7
0 33

340 ALGEBRA
3) S~~a de ~~ ~~~e~~ ~~~i~i~~ ~ ~~~~ ~ega~i~~
Reg~a
Pa~a ~~~a~ ~~ ~~~e~~ ~~~i~i~~ ~ ~~ ~~~e~~ ~ega~i~~
~e ~~~cede a ha~~a~ ~a dife~e~cia a~i~~~~ica de ~~~ ~a~~~e~
ab~~~~~~~ de a~b~~ ~~~e~~~, ~ a~ ~e~~~~ad~ ~b~e~id~ ~e ~e
a~~e~~~e e~ ~ig~~ de~ ~~~e~~ ~a~~~ . C~a~d~ ~~~ d~~ ~~~e-
~~~ ~ie~e~ ig~a~ ~a~~~ ab~~~~~~ ~ ~ig~~~ di~~i~~~~ ~a ~~~a e~
ce~~ . A~~ ~e~e~~~ :
P~de~~~ ~e~~e~e~~a~ ~a ~~~a de ~~ ~~~e~~ ~~~i~i~~ ~ ~~~~ ~ega~i~~ de
~~~ ~ig~ie~~e~ ~~d~~ :
Re~~e~e~~aci~~ g~~fica de ~a ~~~a de ~~ ~~~e~~ ~~~i~i~~ ~ ~~ ~~~e~~
~ega~i~~, e~ ~~e e~ ~~~e~~ ~~~i~i~~ ~ie~e ~a~~~ ~a~~~ ab~~~~~~ ~~e e~ ~ega~i~~ :
3
4
6 -5 -4
i
-3 -2 -1
+6
- 6-
+6
+2
i
FIGURA 4
~ega~i~~, e~ ~~e e~ ~~~e~~ ~ega~i~~ ~ie~e ~a~~~ ~a~~~ ab~~~~~~ ~~e e~ ~~~i~i~~ :
+3 +4 +5
-- 6 ,
' + 2---~
0 +1 +2 +3
-5 -4 -3 -2 -1
11
FIGURA 5
~ega~i~~, e~ ~~e e~ ~a~~~ ab~~~~~~ de a~b~~ ~~~e~~~ e~ ig~a~ .
0
6 >,
6
I
+3 -+4 +5 +6
(-i-6)+(-2)=+4
(-6)+(+2)=-4
(-6)+(+6)=0
(+6)+(-6)=0

4) S~~a c~c cc~O ~ ~~ ~~~'~~~-)~ ~~~i~i~~ ~ ~ega~i~~
Reg~a
La ~~~a de ce~~ c~~ c~a~~~ie~ ~~~e~~ ~~~i~i~~ ~ ~ega~i~~ ~~~ da~~
e~ ~i~~~ ~~~e~~ ~~~i~i~~ ~ ~ega~i~~ .
A~~ ~e~e~~~ :
(+4) +O= + 4
(-4)+0=-4
E~ ge~e~a~ : a + 0 = 0 + a = a
E~ ~~e a ~~ede ~e~ ~~~i~i~~, ~ega~i~~ ~ ~~~~ .
SUSTRACCION DE NUMEROS RELATIVOS
~~e e~ ~~ ~~e ~~e~~a~~~ de~~~~~a~, e~ deci~, ~~e ~a~a ha~~a~ ~a dife~e~cia
e~~~e ~ ~ ~ ba~~a ~~~a~~e a ~ e~ ~~~e~~~ de ~ (~') . Y c~~~ he~~~ ~i~~~ ~~e
~a~a ha~~a~ e~ ~~~e~~~ de ~~ ~~~e~~ ba~~a ca~bia~~e e~ ~ig~~, ~~de~~~ e~~~-
cia~ ~a ~ig~ie~~e
REPRESENTACION GR~FICA DE LA SUSTRACCION DE NUMEROS RELATIVOS
P~~ ~edi~ de ~a i~~e~~~e~aci~~ ge~~~~~ica de ~a ~~~~~acci~~ de ~~~e~~~
~e~a~i~~~, ~~de~~~ e~~~e~a~ ~a di~~a~cia, e~ ~~idade~, ~~e ha~ e~~~e e~ ~~~~~
~~e ~e~~e~e~~a a~ ~i~~e~d~ ~ e~ ~~~~~ ~~e ~e~~e~e~~a a~ ~~~~~ae~d~, a~~ c~~~
e~ ~e~~id~ (~ega~i~~ ~ ~~~i~i~~) de e~a di~~a~cia .
035
M1
Reg~a
Pa~a ha~~a~ ~a dife~e~cia e~~~e d~~ ~~- (+8)-(+4)=(+8)+(-4)=+4
~e~~~ ~e~a~i~~~ ~e ~~~a a~ ~i~~e~d~ e~ ~~~- (+8)-(-4)=(+8)+(+4)=+12
~~ae~d~, ca~bi~~d~~e e~ ~ig~~ . (-8)-(+4)=(-8)+(-4)=-12
A~~ : __1 ,111
(-8)-(-4)=(-8)+(+4)=-4
L~a~a~~~ ~~~e~~~ de ~~ ~~~e~~ a~ ~i~~~ ~~~e~~ c~~
~ig~~ c~~~~a~i~ . A~~, deci~~~ ~~e - ~ e~ ~~~e~~~ de + ~ .
Ya ~i~~~ e~ ~~ ca~~ de ~a ~~~a ~~e : T
(+ ~) + (- ~) = 0
La ~~~~~acci~~ e~ ~~a ~~e~aci~~ i~~e~~a de ~a ~~~a
c~~~i~~e e~ ha~~a~ ~~ ~~~e~~ ~ (~~a~ad~ dife~e~cia), ~a~
~~e
~~e,
~ + ~ = ~ (1)
~~~ad~ c~~ ~~ ~~~e~~ dad~ ~, d~ ~~ ~e~~~~ad~ ig~a~ a
~~~e~~ ~. de ~~d~ ~~e ~e ~e~ifi~~e :
~~~~
1
L~a~a~d~ ~' a~ ~~~e~~~ de ~, ~~de~~~ de~e~~i~a~
~a dife~e~cia ~, ~~~a~d~ e~ a~b~~ ~ie~b~~~ de ~a
~ + ~ + ~' - ~ + ~'
- (2)
ig~a~dad (1), e~ ~~~e~~ ~' ; e~ efec~~ :
(3)
Si ~b~e~~a~~~ e~ ~~i~e~ ~ie~b~~ de e~~a ig~a~dad (2), ~ = ~ + ~'
~e~e~~~ ~~e a~~ica~d~ e~ a~i~~a de a~~cia~i~idad ~e~e~~~ :
~~ + ~' 0, ~ c~~~ ~ + 0 = ~, ~e~d~e~~~ :
T

36 ALGEBRA
Pa~a e~~~e~a~ ~a dife~e~cia (+ 4) - (- 8) = + 12, ~e~d~e~~~ :
~
-0
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4
FIGURA 7
Pa~a e~~~e~a~ ~a dife~e~cia (- 8) - (+ 4) _ - 12, ~e~d~e~~~ :
MULTIPLICACION DE NUMEROS RELATIVOS
Reg~a
E~ ~~~d~c~~ de d~~ ~~~e~~~ ~e~a~i~~~ ~e ha~~a ~~~~i~~ica~d~ ~~~ ~a~~~e~
ab~~~~~~~ de a~b~~ . E~ ~~~d~c~~ ha~~ad~ ~~e~a~~ ~ig~~ ~~~i~i~~ (+), ~i ~~~
~ig~~~ de a~b~~ fac~~~e~ ~~~ ig~a~e~ ; ~~e~a~~ ~ig~~ ~ega~i~~ (-), ~i ~~~ fac-
~~~e~ ~ie~e~ ~ig~~~ di~~i~~~~ . Si ~~~ de ~~~ fac~~~e~ e~ 0 e~ ~~~d~c~~ ~e~~ 0 .
+12
-12
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4
C~a~d~ ~~e~a~~~ c~~ ~~~b~~~~ ~i~e~a~e~
e~ ~~~d~c~~ e~ ~ie~~~e i~dicad~, bie~ e~ ~a
f~~~a a ~ b ; bie~ e~ ~a f~~~a a . b ; ~ ~~~
~~~a~~e~~e ab .
A~~ : i
E~ ~ig~ie~~e c~ad~~ e~ ~~ ~edi~ de ~e- + ~~~ + da + + ~~~ - da -
c~~da~ f~ci~~e~~e ~a ~e~ de ~~~ ~ig~~~ e~ ~a - ~~~ - da + - ~~~ + da -
~~~~i~~icaci~~ de ~~~ ~~~e~~~ ~e~a~i~~~ . ,/'
REPRESENTACION GRAFICA DEL PRODUCTO DE DOS NUMEROS RELATIVOS
E~ ~~~d~c~~ de d~~ ~~~e~~~ ~e~a~i~~~ ~~ede e~~~e~a~~e ge~~~~~ica~e~~e
c~~~ e~ ~~ea de ~~ ~ec~~~g~~~ c~~~ ~a~g~ ~ c~~~ a~ch~ ~ie~e~ dad~~ ~~~
a~b~~ ~~~e~~~ . A e~~a ~~ea ~~de~~~ a~~ib~i~~e ~~ ~a~~~ ~~~i~i~~ ~ ~ega~i~~,
(+2) (+3)=+6 (0) (+3)=0
(-2) (-3)=+6 (0) (-3)=0
(+2) (-3)=-6 00=0
(-2) (+3)=-6

~eg~~ ~~e ~~~ ~ad~~ ~e~ga~ ~a~~~e~ de ~~ ~i~~~ ~e~~id~ ~ de ~e~~id~~ di~-
~i~~~~ ~e~~ec~i~a~e~~e .
6
A
E
-3
3
+6
NOTAS SOBRE EL CONCEPTO DE NUMERO ~ 3 7
+2 +2
FIGURA 9 1
POTENCIA DE NUMERO$ RELATIVOS
L~a~a~~~ ~~~e~cia de ~~ ~~~e~~ ~e~a~i~~ a~ ~~~d~c~~
de ~~~a~~~ c~~~ fac~~~ ~a~~a~ ~ece~ c~~~ ~e ~~ie~a . Si a
e~ ~~ ~~~e~~ ~e~a~i~~ c~a~~~ie~a ~ ~ > 1 e~ ~~ ~~~e~~ ~a c
~a~~~a~, ~e~d~e~~~ ~a ~~~aci~~ a~, ~~e ~e ~ee a e~e~ad~ a ~a a~=a .a .a a
e~~~i~a ~~~e~cia . e i~dica ~~e a debe ~~~a~~e c~~~ fac~~~ ~
~ece~ . A~~ :
E~ ~a ~~~aci~~ a~ = ~, ~~a~a~~~ ~~~e~cia a~ ~~~d~c~~ ~, ba~e a~
~~~e~~ ~~e ~~~a~~~ c~~~ fac~~~ a, ~ e~~~~e~~e a ~, ~~e ~~~ i~dica
~a~ ~ece~ ~~e debe~~~ ~~~a~ c~~~ fac~~~ a a . A ~a ~~e~aci~~ de ha~~a~
e~ ~~~d~c~~ ~, ~a ~~a~a~~~ ~~~e~ciaci~~ ~ e~e~aci~~ a ~~~e~cia .
E~e~~~~ :
2
i
+6
+3
+3
~
-6
E~ e~~e e~e~~~~, 4 e~ ~a ba~e ; 5 e~ e~ e~~~~e~~e, ~ 1024 e~ ~a ~~~e~cia .
Reg~a
La ~~~e~cia de ~~ ~~~e~~ ~~~i~i~~ ~ie~~~e e~ ~~~i~i~a . La ~~
~e~cia de ~~ ~~~e~~ ~ega~i~~ ~e~~ ~~~i~i~a ~i e~ e~~~~e~~e e~ e~~e~~
~ ~a~ : ~ega~i~a ~i c~ e~~~~e~~e e~~e~~ e~ i~~a~ . A~~ :
4 5 = 1024

380 ALGEBRA
PRODUCTO DE DOS POTENCIAS DE IGUAL BASE
Reg~a
Pa~a ~~~~i~~ica~ d~~ ~~~e~cia~ de ig~a~ ba~e,
~e e~e~a dicha ba~e a ~a ~~~e~cia ~~e ~e~~~~e de ~a
~~~a de ~~~ e~~~~e~~e~ ~e~~ec~i~~~ . E~e~~~~ :
POTENCIA DE UNA POTENCIA
Reg~a
Pa~a ha~~a~ ~a ~~~e~cia de ~~a ~~~e~cia ~e ~~~-
~i~~ica~ ~~~ e~~~~e~~e~ ~ ~e ~a~~ie~e ~a ba~e ~~i~i-
~i~a .
~i~a . E~e~~~~ :
Ha~ ~~e ~~~e~ e~~ecia~ c~idad~ e~ ~~ c~~f~~-
di~ ~a ~~~e~cia de ~~a ~~~e~cia, c~~ ~a e~e~aci~~ de
~~ ~~~e~~ a ~~a ~~~e~cia c~~~ e~~~~e~~e, a ~a ~e~
e~~~ afec~ad~ ~~~ ~~~~ e~~~~e~~e . A~~, ~~ e~ ~~ ~i~~~
(4 2)3 ~~e (4 23 ) . E~e~~~~ : %`
a~ . a ~ = a ~+~
(3) 2 (3) 4 = 32+4 = 3 0 = 729
(a~~)"' = a ~~~ = a~-
22)3 = -2 2~3 =-2 6 -64
(42)8 = 42~8 = 4 0 = 4096
(42 3 ) = 42~2 .2 = 4 8 = 65536
DIVISION DE NUMEROS RELATIVOS
Ya ~i~~~, a~ ~~a~a~ de ~a~ ~e~e~ f~~~a~e~ de ~a ~~~~i~~icaci~~, ~~e de
ac~e~d~ c~~ e~ a~i~~a VI (e~i~~e~cia de~ i~~e~~~), a ~~d~ ~~~e~~ ~ea~ a # 0,
c~~~e~~~~de ~~ ~~~e~~ ~ea~, ~ ~~~~ ~~~, ~, de ~~d~ ~~e a~ = 1 : E~~e ~~-
~e~~ ~ ~e ~~a~a i~~e~~~ ~ ~ec~~~~c~ de a, ~ ~e ~e~~e~e~~a ~~~ 1/a .
E~ i~~e~~~ de -f 4 e~ + 4 1
E~ i~~e~~~ ~ ~ec~~~~c~ de ~~ ~~~e~~ ~e~a- E~ i~~e~~~ de - 4 e~ --1
~i~~ c~a~~~ie~a di~~i~~~ de ce~~ ~ie~e ~~ ~i~~~
E~ i~~e~~~ de - 4e e~
., '
~ig~~ . , 3
E~ i~~e~~~ de + 1 e~ + 2
La di~i~i~~ e~ ~~a ~~e~aci~~ i~~e~~a de ~a ~~~~i~~icaci~~ ~~e c~~~i~~e
e~ ha~~a~ ~~~ de ~~~ fac~~~e~, c~~~cid~~ e~ ~~~~ fac~~~ ~ e~ ~~~d~c~~ . E~ deci~,
dad~ e~ di~ide~d~ d ~ e~ di~i~~~ d' ha~~a~ e~ c~cie~~e c, de ~~~d~ ~~e ~e ~e-
~ifi~~e d'c = d .
Rec~~da~~~ ~~e e~~a ~~e~aci~~ ~~~~ e~ ~~~ib~e ~i d' e~ di~~i~~~ de ce~~ .
A~~ica~d~ e~ a~i~~a de e~i~~e~cia de~ i~~e~~~, ~e~e~~~ ~~e :
De ~~ c~a~ ded~ci~~~ ~a ~ig~ie~~e
Reg~a
Pa~a di~idi~ ~~ ~~~e~~ c~a~~~ie~a d ~~~ ~~~~ ~~~e~~ di~~i~~~ de ce~~ d',
~~~~i~~ica~~~ d ~~~ e~ ~ec~~~~c~ d' (1/d') . E~ c~cie~~e ~~e ~e~~~~e ~e~~ ~~~i~i~~
~i ~~~ d~~ ~~~e~~~ ~~~ de~ ~i~~~ ~ig~~ ; ~ ~ega~i~~, ~i ~~~ de ~ig~~~ c~~~~a~i~~ .
+ e~~~e + (~a +
C~~ e~ ~ig~ie~~e c~ad~~ ~~de~~~ ~ec~~da~ f~ci~~e~~e ~a - e~~~e - (~a +
~e~ de ~~~ ~ig~~~ de ~a di~i~i~~ c~~ ~~~e~~~ ~e~a~i~~~ . / + e~~~e - da -
- e~~~e + (~a -
Sabe~~~ ~~e :
1/d' (d'c) = 1/d' d
1/d' (d'c) = (1/d' d') c = (+ 1) c = c
E~i~i~a~d~ ~~eda : c = 1/d' d

Ah~~a ~~e e~~~dia~~~ ~a di~i~i~~, ~~de~~~ e~~~cia~ ~~e~ ca~~~ de ~a
3) La di~i~i~~ de d~~ ~~~e~cia~ de ig~a~ ba~e e~ ig~a~
a ~a ba~e e~e~ada a ~a ~~~e~cia ~~e d~ ~a dife~e~cia de a~b~~
e~~~~e~~e~ . A~~ : ---
UNIFORMIDAD DE LAS OPERACIONES FUNDAMENTALES CON NUMEROS RELATIVOS
POSIBILIDAD DE AMPLIAR EL CAMPO NUMERICO
L~~ ~~~e~~~ ~ea~e~ ~~ cie~~a~ ~a ~~~ibi~idad de a~~~iaci~~ de~ ca~~~
~~~~~ic~ . Ta~ ~~~ibi~idad ~e ~a~~ie~e abie~~a ~a~a ~a i~~~~d~cci~~ de ~~e~~~
e~~e~, ~ie~~~e ~~e ~a~e~ e~~e~ c~~~~a~ ~a~ ~e~e~ f~~~a~e~ . De~~~~ de ~~~ ~~~i~e~
de e~~e ~e~~~, e~ e~~~dia~~e ~~da~~a ~e e~f~e~~a~~ c~~ ~~a ~~e~a a~~~iaci~~
de~ ca~~~ ~~~~~ic~ . Se ~~a~a de~ ~~~e~~ c~~~~e~~, ~~e e~ ~~ ~a~ de ~~~e~~~
dad~~ e~ ~~ ~~de~ de~e~~i~ad~ ~ ~~e e~~~ c~~~~i~~id~ ~~~ ~~ ~~~e~~ ~ea~
~ ~~ ~~~e~~ i~agi~a~i~, C~~ e~~~~ ~~~e~~~ ~~d~e~~~ ~e~~e~e~~a~ ~~ ~~~~~
c~a~~~ie~a e~ e~ ~~a~~ . E~ e~ ca~~~~~~ XXXII ~e ~~e~e~~a~~ ~~a di~c~~i~~
a~~~ia ~~b~e e~~~~ ~~~e~~~ .
0 3 9
He~~~ ~i~~~ e~ ~a~ ~~e~aci~~e~ e~~~diada~, a ~abe~ : ~~~a, ~e~~a, ~~~~i~~i-
caci~~, ~~~e~ciaci~~ ~ di~i~i~~, ~~e ~e c~~~~e e~ ~~da~ e~~a~ e~ a~i~~a (~e
~~if~~~idad . Q~ie~e e~~~ ~ig~ifica~ ~~e c~a~d~ ~~~e~e~~~~ d~~ ~~~e~~~ ~e~a-
~i~~~ a c~a~~~ie~a de ~a~ ~~e~aci~~e~ ~e~ci~~ada~, e~ ~e~~~~ad~ e~ ~~~, ~ ~~~~
~~~, e~ deci~, ~~ic~ . Si~ e~ba~g~, c~a~d~ e~~~ae~~~ ~a ~a~~ c~ad~ada de ~~
~~~e~~ ~~~i~i~~, ~e~e~~~ ~~ ~e~~~~ad~ d~b~e . P~e~ c~~~ ~e~e~~~, a~ e~~~dia~
~a e~~~acci~~ (~e ~a~ ~a~ce~, ~~ ~~~e~~ ~~~i~i~~ c~a~~~ie~a ~ie~~~e ~ie~e d~~
~a~ce~ de g~ad~ ~a~,~~a ~~~i~i~a ~ ~~~a ~ega~i~a .
A~~ : f+ aa = --* a' ~~~~~e : (+ a') 2 = (+ a') (+ a') = + a
(-a')2=(- a') (- a') = + a
de~ ~i~~~ ~~d~ : /+ 64 = ~ 8 ~~~~~e : (+ 8) 2 = (+ 8) (+ 8) = + 64
(- 8) 2 = 1- 8) (- 8) = + 64
3 4 =3 4-2 =3 2 =9
3 2
3-2 = 1 1
32
9
a-
- = a ~ -~
a~
e~e~aci~~ a ~~~e~cia de ~~ ~~~e~~ c~a~~~ie~a .
1) Si ~~ ~~~e~~ c~a~~~ie~a a=91=0, ~e
a~ =+1
e~e~a a ~a ~~~e~cia 0 e~ ig~a~ a + 1 . A~~ : / 30 =+1
2) Si ~~ ~~~e~~ c~a~~~ie~a a =A0, ~e e~e~a a ~~ e~~~~e~~e 1
~ega~i~~ c~a~~~ie~a -7~ e~ ig~a~ a~ ~ec~~~~c~ de ~a ~~~e~cia a ~", de
e~~~~e~~e ~~~i~i~~ . A~~ :
a
- a~

EL ALC,EBRA EN EL ANTIGUO EGIPTO (5,000-500
A . C .) E~ Egi~~~, ~a~a~i~~~~~ ~~eb~~ de fa~a~~e~ ~
~i~~~ide~, e~c~~~~a~~~ ~~~ ~~i~e~~~ ~e~~igi~~ de~ de-
~a~~~~~~ de ~~a cie~cia ~a~e~~~ica . S~~ e~ige~cia~ ~i-
~a~e~, ~~~e~a~ a ~a~ ~e~i~dica~ i~~~daci~~e~ de~ Ni~~,
SUMA
33 LA SUMA O ADICION e~ ~~a ~~e~aci~~ ~~e ~ie~e ~~~ ~b~e~~ ~e~~i~
d~~ ~ ~~~ e~~~e~i~~e~ a~geb~aica~ (~~~a~d~~) e~ ~~a ~~~a e~~~e~i~~
a~geb~aica (~~~a) .
A~~, ~a ~~~a de a ~ b e~ a + b, ~~~~~e e~~a ~~~i~a e~~~e~i~~ e~ ~a ~e~-
~i~~ de ~a~ d~~ e~~~e~i~~e~ a~geb~aica~ dada~ : a ~ b .
La ~~~a de a ~ - b e~ a - b, ~~~~~e e~~a ~~~i~a e~~~e~i~~ e~ ~a
~e~~i~~ de ~a~ d~~ e~~~e~i~~e~ dada~ : a ~ - h .
CAR~CTER GENERAL DE LA SUMA ALGEBRAICA
E~ A~i~~~~ica, ~a ~~~a ~ie~~~e ~ig~ifica a~~e~~~, ~e~~ e~ A~geb~a
~a ~~~a e~ ~~ c~~ce~~~ ~~~ ge~e~a~, ~~e~ ~~ede ~ig~ifica~ a~~e~~~ ~ di~-
~>~i~~ci~~, ~a ~~e ha~ ~~~a~ a~geb~aica~ c~~~ ~a de~ ~~~i~~ e~e~~~~, ~~e
e~~i~a~e a ~~a ~e~~a e~ A~i~~~~ica .
Re~~~~a, ~~e~, ~~e ~~~a~ ~~a ca~~idad ~ega~i~a e~~i~a~e a ~e~~a~ ~~a
ca~~idad ~~~i~i~a de ig~a~ ~a~~~ ab~~~~~~ .
A~~, ~a ~~~a de ~ ~ - ~ e~ ~ - ~, ~~e e~~i~a~e a ~e~~a~ de ~ e~ ~a~~~
ab~~~~~~ de - ~ ~~e e~ ~~i .
La ~~~a de - 2~ ~ - 3~ e~ - 2~ - 3~, ~~e e~~i~a~e a ~e~~a~ de - 2~ e~
~a~~~ ab~~~~~~ de - 3~ ~~e e~ 13~J .
40
~~~ ~~e~a~~~ a ~e~fecci~~a~ ~a A~i~~~~ica ~ ~a Ge~~e-
~~~a . E~ e~ ~a~i~~ de Rhi~d, debid~ a~ e~c~iba Ah~e~
(1650 A . C .), e~ ~~~ ~a~i~~~ ~ a~~ig~~ d~c~~e~~~
~a~e~~~ic~ ~~e e~i~~e, ~e ~~e~e~~a~ e~~~e ~~~~i~~e~
~~~b~e~a~, ~~~~ci~~e~ de ec~aci~~e~ de ~eg~~d~ g~ad~,
CAPITULO

35 REGLA GENERAL PARA SUMAR
Pa~a ~~~a~ d~~ ~ ~~~ e~~~e~i~~e~ a~geb~aica~ ~e e~c~ibe~ ~~a~ a c~~-
~i~~aci~~ de ~a~ ~~~a~ c~~ ~~~ ~~~~i~~ ~ig~~~ ~ ~e ~ed~ce~ ~~~ ~~~~i~~~ ~e-
~e~a~~e~ ~i ~~~ ha~ .
I . SUMA DE MONOMIOS
1) S~~a~ 5a, 6b ~ 8c .
L~~ e~c~ibi~~~ ~~~~ a c~~~i~~aci~~ de ~~~~~ c~~ ~~~ 5a + 6b + 8c . R .
~~~~i~~ ~ig~~~, ~ c~~~ 5a=+5a, 6b=+6b ~ 8c=+8c ~a ~~~a ~e~~ : ~
E~ ~~de~ de ~~~ ~~~a~d~~ ~~ a~~e~a ~a ~~~a . A~~, 5a + (ib + 8c e~ ~~
~i~~~ ~~e 5a + 8c + 6b ~ ~~e 6b + 8c + 5a .
E~~a e~ ~a Le~ C~~~~~a~i~a de ~a ~~~a .
2) S~~a~ 3a 2 b, 4ab 2 , a 2 b, 7ab 2 ~ 6b 3 .
Te~d~e~~~ :
3a '-'b + 4ab 2 + a 2 b + 7ab 2 + 6b 3 .
Red~cie~d~ ~~~ ~~~~i~~~ 4a 2 b + ~~ab 2 + 6b 3 . R .
~e~e~a~~e~, ~~eda : -
3) S~~a~ 3a ~ - 2b .
C~a~d~ a~g~~ ~~~a~d~ e~ ~ega~i~~, ~~e~e i~c~~i~~e 3a + (- 2b)
de~~~~ de ~~ ~a~~~~e~i~ ~a~a i~dica~ ~a ~~~a ; a~~ : .
La ~~~a ~e~~ :
`3a
-
2b R
4) S~~a 7a, - 8b, - 15a, 9b, - 4c ~ 8 .
Te~d~e~~~ :
7a+(-8b)+(-15a)+9b+(-4c .)+8=7a-8b-15a+9b-4c+8=-8a+b-4c+8 . R .
5) S~~a~ ?d~, ~ab, -2b', - 8ab, 3a 2 , - g b 2 .
2 a 2 + ~ab + (- 2b 2 ) + (- 3 ab) +!a 2 + (-
:S
$2 )
3 2 i S
b
=
~
a 2 + - ab - 21)* - ~ab + 3a" - -b2 = a 2 - ~ab - g b 2 . R .
EJERCICIO 15
S~~a~ :
1 . ~, ~ . 11 . -11 ~, 8~ .
2 . ~, -~ . 12 . 9ab, -15ab .
3 . -3a, 4b . 13 . -~~, -9~~ .
4 . 5b, -6a . 14 . i~~, -~~~~ .
5 . 7, -6 .
6 . -6, 9 . 15 .
7 . -2~, 3 ~.
8 . 5~~, -~ . 16 . ~
-b,
~
-c .
9 . 5a, 7a .
10 . -8~, -5~ . 17 .
f
1
~a,-
2
ab
.
3 b, ~b .
21 .
22 .
23 .
18 . - ~
~~, - 2
~~.
19 . - ~
abc, -
~
abc .
20 . -4~ 2 ~, ~~2 ~ .
3 8
-~~, --i~~ .
~ 4
a, b, c .
a, - b, c .
SUMA
~ 41
24 . a, -b, 2c .
25 . 3~, -2~, 4~ .
26 . a 2 , - 7ab, -5b 2.
27 . X2, -3~~, -4~2 .
28 . X3, -~ 2~, 6 .
29 . 2a, -b, 3a .
30 . -i~, -8~, 4~ .
31 . -7a ; 8a, -b-
1 2 8
32 . 2_~, $ ~, - 4~ .

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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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~~

RESTA ~ 4 7
2) Re~~a~ 4b de 2a .
E~c~ibi~~~~ e~ ~i~~e~d~ 2a c~~ ~~ ~ig~~ ~ a c~~~i~~a- 2a-4b . R .
ci~~ e~ ~~~~~ae~d~ 4b c~~ e~ ~ig~~ ca~biad~ ~ ~a ~e~~a ~e~~ :
E~ efec~~ : 2a-4b e~ ~a dife~e~cia, ~~~~~e ~~- 2a - 4b + 4b = 2a .
~~ada c~~ e~ ~~~~~ae~d~ 4b ~e~~~d~ce e~ ~i~~e~d~ :__ /
3) Re~~a~ 4a 2 b de - 5a 2 b .
E~c~ib~ e~ ~i~~e~d~ - 5a 2 b ~
-5a ~ b -4a ~b 9a ~b
a c~~~i~~aci~~ e~ ~~~~~ae~d~ 4a 2b
= - . R .
c~~ e~ ~ig~~ ca~biad~ ~ ~e~g~ : %
- 9a~b e~ ~a dife~e~cia, ~~~~~e ~~~ada c~~ -9a 2b + 4a 2 b = - 5a 2 b .
e~ ~~~~~ae~d~ 4(~~b ~e~~~d~ce e~ ~i~~e~d~ :
4) De 7 ~e~~a~ - 4 .
C~a~d~ e~ ~~~~~ae~d~ e~ ~ega~i~~ ~~e~e i~c~~i~~e de~-
~~~ (~e ~-~~ ~a~~~~e~i~ ~a~a i~dica~ ~a ~~e~aci~~, de e~~e ~~-
7-
d~ di~~i~g~i~~~ e~ ~ig~~ - ~~e i~dica ~a ~e~~a de~ ~ig~~ - (- 4)=7+4=11 . R .
~~e ~e~a~a e~ ca~~c~e~ ~ega~i~~ de~ ~~~~~ae~d~ . A~~ : '
E~ ~ig~~ - de~a~~e de~ ~a~~~~e~i~ e~~~ ~a~a i~dica~ ~a ~e~~a ~ e~~e ~ig-
~~ ~~ ~ie~e ~~~ ~b~e~~ ~~e deci~~~~, de ac~e~d~ c~~ ~a ~eg~a ge~e~a~ ~a~a
~e~~a~, ~~e debe~~~ ca~bia~ e~ ~ig~~ a~ ~~~~~ae~d~ - 4 . P~~ e~~ - a c~~~i-
~~aci~~ de~ ~i~~c~~~ 7 e~c~ibi~~~ +4 .
5) De 7~ 3 ~' ~e~~a~ - 8~ 3 1ia
Te~d~e~~~ : 7~3 ~4 - (- 8~ 3 ~ 4 ) = 7~ 3 ~' + 8~ 3 ~ 4 =15~ 3 ~' . R .
6) De - i ab ~e~~a~ - i ab .
Te~d~e~~~ : -1 ab - (-1 ab) ab . R .
= - ab + 1 ab =
CAR~CTER GENERAL DE LA RESTA ALGEBRAICA
E~ A~i~~~~ica ~a ~e~~a ~ie~~~e i~~~ica di~~i~~ci~~, ~ie~~~a~ ~~e ~a
~e~~a
~~e
~a~e
~i~~ci~~
Ha~
a ~~~a~
a~geb~aica
~a dife~e~cia
L~~ e~e~~~~~
~ a~~e~~~ .
~e~~a~
~a
~ie~e ~~ ca~~c~e~
a~geb~aica~,
e~ ~a~~~
4, 5 ~ 6 ~~~
~i~~a ca~~idad
c~~~
~~e e~
dice~
~~~ ge~e~a~,
~a~ de ~~~
~i~~e~d~ .
~~e ~e~~a~
~~~i~i~a .
~~e~ ~~ede
e~e~~~~~
~~a ca~~idad
~ig~ifica~
4 ~ 5 a~~e~i~~e~,
~ega~i~a
di~-
e~
e~~i-
EJERCICIO 20
De :
1 . -8 ~e~~a~ 5 . 6 . 2a ~e~~a~ 3b . 11 . -9a 2 ~e~~a~ 5b 2 .
2 . -7 ~ 4 . 7 . 3b ~ 2 . 12 . -7~~ ~ -5~~ .
3 . 8 ~ 11 . 8 . 4~ ~ 6b . 13 . 3a ~ 4a .
4 . -8 -11 . 9 . -5a 6b . 14 . 11 ~2 ~ 2,5 ~2
5 . -1 11 -9 . 10 . -8~ ~ -3 . 15 . -6~ 2 ~ 11 -~ ~~ .

II . RESTA DE POLINOMIOS
41 C~a~d~ e~ ~~~~~ae~d~ e~ ~~ ~~~i~~~i~, ha~ ~~e ~e~~a~ de~ ~i~~e~d~
cada ~~~ de ~~~ ~~~~i~~~ de~ ~~~~~ae~d~, a~~ ~~e a c~~~i~~aci~~ de~
~i~~e~d~ e~c~ibi~e~~~ e~ ~~~~~ae~d~ ca~bi~~d~~e e~ ~ig~~ a ~~d~~ ~~~
~~~~i~~~ .
E~e~~~~~
(1) De 4~ - 3~ + ~ ~e~~a~ 2~ +5~-6 .
La ~~~~~acci~~ ~e i~dica i~c~~~e~d~ e~ ~~~~~ae~- 4~ - 3~ + ~ - (2~ + S~ - 6) .
d~ e~ ~~ ~a~~~~e~i~ ~~ecedid~ de~ ~ig~~ -, a~~ :
Ah~~a, de~a~~~ e~ ~i~~e~d~ c~~ ~~~ ~~~~i~~ ~ig-
~~~ ~ a c~~~i~~aci~~ e~c~ibi~~~ e~ ~~~~~ae~d~ 4~ - 3~ + ~ - 2~ - 5~ + 6 .
ca~bi~~d~~e e~ ~ig~~ a ~~d~~ ~~~ ~~~~i~~~ ~ ~e~-
d~e~~~ :
Red~cie~d~ ~~~ ~~~~i~~~ ~e~e~a~~e~, ~e~d~e~~~ : , 2~-3~-4~+6 . R .
E~ ~a ~~~c~ica ~~e~e e~c~ibi~~e e~ ~~~~~ae~d~ c~~ ~~~ ~ig~~~ ca~biad~~ deba-
~~ de~ ~i~~e~d~, de ~~d~ ~~e ~~~ ~~~~i~~~ ~e~e~a~~e~ ~~ede~ e~ c~~~~~a ~
~e hace ~a ~ed~cci~~ de ~~~~~, ~e~a~~~d~~~~ ~~~~ de ~~~~~ c~~ ~~~ ~~~~i~~ ~ig~~~ .
4~-3~+ ~
A~~, ~a ~e~~a a~~e~i~~ ~e ~e~ifica de e~~a ~a~e~a : ---' - 2~ - 5~ + 6
2~-3~-4~+6 . R .
48 ALGEBRA
16 . 11a 3 ~2 ~e~~a~ -7a 3 ~2 . 22 . 6a~ ~e~~a~ -5a" . 27 . - 2
~e~~a~
3
17 . -8ab 2 ~ -8ab 2 . 23 . -45a~ -1 ~ -60a ~-1 .
3
1
4
18 . 28 ~ ~
~
-- -
2
--~ 2 .
31~ 2 ~ -46~-'~ . 24 . 54b~ -1 ~ - 86 b ~-1 3 3
19 . -84a 2 b -84a 2 b 4
20 . 3a~+ 1
11
26 . -35~" , . -60~" . 29 . ~3~ ~ _ 5 ~3~,
5b~ , 2 .
~
1
11 .
21 . -8~a+ 2 ~ 26 . 5 ~ 30 . _ Iab 2 -
3
ab 2 .
8 4
31 . 3
Re~~a~
de -2 . 43. -a de 3a . de -85a~ + 2_
55 . 54a' + 2
32 -1 7 . 44 . -3b -4b .
33 . -5
~
~ -8 . 45 . -11~ 3
~
~ 54~ 3 . 56 . -6a 1
34 . -4 ~ 5 . 46 . 14a 2 b 78a 2 b .
2
35 . -7
36. -5
~ -7 .
2a .
47 . -43a-~ -
48 . 9ab
~
~
-54a 2 ~.
-ab .
57 . -5 - 3 .
37 . b -3~ . 49 . -31 7
, . -31~ 2 ~ ~2 ~. 58 . g ~~ a
-~3 .
38 . 5~ ~ -2~ . 50 . a~ ~ -3a~ . - ~ 10
39 . -6a 3b . 51 . -7a~+ 1 ~a~ I1 .
31
40 . -5a 3
11
8b . 52 . !)~~
11
105W 59 . -1- a 2 b 2 ~
~
-a'-6 2.
41 . -9
~
~ 53 . 18a~-1
~
-31a~ -1 .
12 ~
-7a .
42 . -25 ~ 25ab . 54 . -19~ ~
~
~ -236?0 . 60 . 45a 3 b 2 21
1 a 3 b 2 .
~

1 .
2 .
3 .
4 .
5 .
6 .
7 .
8 .
PRUEBA
La dife~e~cia ~~~ada c~~ e~ ~~~~~ae~d~ debe da~ e~ ~i~~e~d~ .
2~-3~-4~+6
2~ +5~-6
4~-3~+ ~ (~i~~e~d~) .
(2) Re~~a~ - 4a 5 b - ab 5 + 6a 3 b 3 - a"b 4 - 3b~ de 804 b 2 + a~ - 4a"b' + 6ab ~' .
A~ e~c~ibi~ e~ ~~~~~ae~d~, c~~ ~~~ ~ig~~~ ca~biad~~, deba~~ de~ ~i~~e~d~,
debe~ ~~de~a~~e a~b~~ c~~ ~e~aci~~ a ~~a ~i~~a ~e~~a .
A~~, e~ e~~e ca~~, ~~de~a~- a~ + 8a 4 b 2 - 4a 2 b 4 + 6ab 5
d~ e~ ~~de~ de~ce~de~~e + 4a 5 'b - 6a 3 b 3 + a 2 b 4 + ab 5 + 3be
c~~ ~e~aci~~ a ~a a ~e~-
d~e~~~ :----
~a dife~e~cia ~~~a-
da c~~ e~ ~~~~~ae~-
d~, debe da~~~~ e~
~i~~e~d~ :
E~ e~ e~e~~~~ a~~e~i~~, ~~~a~d~ ~a dife-
~e~cia 2~ - 3~ - 4~ + 6 c~~ e~ ~~~~~ae~-
d~ 2~ + 5~ - 6, ~e~d~e~~~ :
(3) Re~~a~ - 8a 2 ~ + 6 - 5a~ 2 - ~3 de 7a 3 + 8a 2 ~ + 7a~'` - 4 ~ ~~~ba~ e~ ~e~~~-
~ad~ ~~~ e~ ~a~~~ ~~~~~ic~ .
7a~ 2 + 8a 2 ~ + 7a 3 - 4
Efec~~e~~~ ~a ~e~~a ~~de~a~d~ c~~ ~e~aci~~ ~8 + 5a~ 2 + 8a 2 ~ - 6
a ~a ~ :
~3 + 12a~ 2 + 16a 2 ~ +7a 3 _10 . R .
La ~~~eba de~ ~a~~~ ~~~~~ic~ ~e efec~~a ha~~a~d~ e~ ~a~~~ ~~~~~ic~ de~ ~i-
~~e~d~, de~ ~~~~~ae~d~ c~~ ~~~ ~ig~~~ ca~biad~~ ~ de ~a dife~e~cia ~a~a
~~ ~i~~~ ~a~~~ de ~a~ ~e~~a~ (e~ ~a~~~ de cada ~e~~a ~~ e~c~ge~~~ ~~~~~~~~) .
Red~cie~d~ e~ ~a~~~ ~~~~~ic~ de ~i~~e~d~ ~ ~~~~~ae~d~ c~~ e~ ~ig~~ ca~-
biad~, debe da~~~~ e~ ~a~~~ ~~~~~ic~ de ~a dife~e~cia .
A~~, e~ e~ e~e~~~~ 7a~ 2 + 8a 2 ~ + 7a 3 - 4 = 28 + 16 + 7 - 4 = 47
a~~e~i~~ ~a~a a=1, ~3 + 5a~ 2 + 8a 2 ~ - . 6 = 8 + 20 + 16 - 6 = 38
~ = 2, ~e~d~e~~~ :
~3 +12a~ 2 +16a"~+7a 3 -10 = 8+48+32+7-10=85
M> EJERCICIO 21
De :
a-I-b ~e~~a~ a-b .
2~-3~ ~e~~a~ -~+2~ .
8a+b ~e~~a~ -3a+4 .
~2 -3~ ~e~~a~ -5~+6 .
a 3 -a'-'b ~e~~a~ 7a 2 b+9ab 2 .
~- ~+~ ~e~~a~ ~- ~+~ .
~+~-~ ~e~~a~ -~-~+~ .
~2+~ 2 -3~~ ~e~~a~ -~ 2 +3~ 2 -4~~ .
RESTA
~6 + 4a-_'b + 8a 4 b 2 -6a 3 b 3 - 3a 2 b 4 + 7ab 5 + 3b~ . R .
a 6 + 4a 5 b + 8a 4 b 2 - 6a 3 b 3 - 3a 2 b 4 + 7ab 5 + 3be
- 4a-'b + 6a 3b3 - a 2 b 4 - ab 5 - 3b~
ae + 8a''b 2 - 4a 2 b 4 + 6ab 5 (~i~~e~d~) .
9 . ~ 3 -~ 2 +6 ~e~~a~ 5~' 2 -4~+6 .
10 . ~2 +6~ :1 -8 ~e~~a~ 2~'-3~-+6~ .
11 . a :'--6ah 2 +9a ~e~~a~ 15a 2 b-8a+5 .
12 . ~4 +9~~ 3 -11~ 4 ~e~~a~ -S~ 3 ~-6~ 2 ~"+20~ 4 .
13 . a+b+c-d ~e~~a~ -a-b+c-d .
14 . ab+2ac-3cd-5de ~e~~a~ -4ac+8ab-5cd+5de .
15 . ~3 -9~+6~ 2 -19 ~e~~a~ -11X2 +21~-43+6X 3 .
16 . ~ 9~ :1 +6~ 2 -31 ~e~~a~ -~~~ 4 +31~ 3 -8~2-19~ .
17 . 5~a 3 -9~ 3 +6~"~-8~~" ~e~~a~ 14~~'=-21~~ 2 ~+5~ 3 -18 .
18 . 4~ 3 ~-19~~ 3 +~ 4 -6~ 2 ~2 ~e~~a~ -~ 4 -51~~ 3 -I-32~ 2 ~2 -2 .5~ 3 ~.
19 . ~"+~ 4 ~2 -9~'~ 4 +19 ~e~~a~ -131~ :I~ 3 +16~~~ 5 -3U~ 2 ~4 -61 .
20 . -a 5 b+6a 3 b 3 -18ab 5 +42 ~e~~a~ -Sa~+9b~-11a 4 b 2 -11a 2 b 4 .
~ 49

1 .
2 .
3 .
4 .
5 .
6 .
7 .
8 .
9 .
~~.
5 0 ~ ALGEBRA
21 . 1-~ 2 +~ 4 -~ 3 +3~-6~ 5 ~e~~a~ -~e+8~ 4 -30~ 2 +15~-24 .
22 . -6~ 2 ~3 +8~ 5 -23~ 4 ~+80~ 3 ~2 -18 ~e~~a~ -~ 5 +9~~ 4 +80-21~ 3 ~2 -51~ 4 ~.
23 . M6-8M4~ 2 +21~ 2 ~4 +8-6~~ 5 ~e~~a~ -23~ 5 ~+14~ 8 ~3 -24~~ 5 +8~e-14 .
24 . ~'-8~+16~ 5 -23~ 2 -15 ~e~~a~ -8~ 8 +25~'-30~ 3 +51~-18 .
25 . 9a~-15a 4 b 2 +31a 2 ~4 -b 6 +14 ~e~~a~ 25a 5 b-15a 4b 2 +53a 3 b 3 -9ab 5 +3b 6 .
26. a ~+a~+~-a~ . 2 ~e~~a~ 5a~-6a~+~-a ~+ 2 .
27 . ~ a- ~a - ~+3~~ -2 ~e~~a~ 3~a+ 1 -4~a+5~9 --2 -~-8~a-3 .
28 . a~ + 4 -7a ~+L- 8a ~+6a~ -1 ~e~~a~ -5a ~ + 3 -14a~+ 2 -~~a'~+ 1 -8a~ -1 ,
29 . ~a+ 2 -7~ a +9~~ -1 +25~a -2 ~e~~a~ -11~ 41 +19~5+45~~ -1 +60~a -3 .
30 . ~~ +1- 6~~ -2 +8~~ -3 -19~~ -5 ~e~~a~ Si~ ~+5~~ -2 4-be~ 3 +~ ~-4 +9~ i-5 .
f EJERCICIO 22
Re~~a~ :
a-b de b-a .
~-~ de 2~+3 ~.
-5a+b de -7a+5 .
~2 -5~ de -~ 2 +6 .
~3 -~~ 2 de ~ 2 ~+5~~ 2 .
6a 2 b-8a 3 de 7a 2 b+5ab 2.
a-b+2c de -a+2b-3c .
~-~+~ de -3~+4~+5~ .
-~+~-~ de ~+3~-6~ .
3a 2 +ab-6b 2 de -5b 2 +8ab+a 2 .
~2 -~ 2 -3~~ de -5~ 2 -~ 2 +6~~ .
-~ 3 -~+6 de -8~ 2 +5~-4
9
~3 +14~ 2 +9 de 14~ 2 -8~+16 .
ab-bc+6cd de 8ab+5bc+6cd .
25a 2 b-8ab 2 -b 3 de a- 1 -9a-"b-b 3 .
~~2-6~3+4 de 6~ 3 -8~ ~2 ~-6~~ 2.
~2 +7~-8c+d de ~2 -9~+~~c+14 .
7a 3 b+5ab :I-8a 2 b 2 +b 4 de 5a 4 +9a " b-40ab 3 +6b 4.
6~ 3 -9~+6~ 2 -7 de ~~-8~ 4 +25~ 2 +15 .
~5 -~ 2 ~3 +6~~ 4 +25 ~5 de -3~~ 4 -8~ 3 ~2 -19~ 5 +18 .
11 .
12 .
13 .
14 .
15 .
16 .
17 .
18 .
19 .
2 0 .
1
(4) De 1 ~e~~a~ ~ 2 +~+5 . -5-~-~ 2
-4-~-~ 2 . R .
E~ ~~~~~ae~d~ ~ 2 + ~ + 5 ~~~ad~ c~~ ~a di-
fe~e~cia -- 4 - ~ - ~ 2 ~~~ da e~ ~i~~e~d~ : -
( 5) Re~~a~ 9ab 3 - 11 a 3 b + 8a 2 b 2 - b 4 de a' - 1 .
Te~d~e~~~ : a 4 - 1
~~a 3 b - 8a 2 b 2 - 9ab 3 + b 4
~2 +~+5
-~ 2 -~-4
1 (~i~~e~d~) .
a 4 + ~~a 3 b - 8a 2 b 2 - 9ab 8 + b 4- 1 . R . .
f
1 .
EJERCICIO 23
De :
1 ~e~~a~ a-1 . 3 . -9 ~e~~a~ 3a+a 2 -5 . 5 . 1 ~e~~a~ a 3 -a 2 b+ab 2 .
2 . 0 ~e~~a~ a-8 . 4 . 16 ~e~~a~ 5~~-~ 2 +16 . 6 . ~3 ~e~~a~ -~ 3 -8~ 2 ~-6~~ 2 .
21 . 25~+25~ 3 -18~ 2 -11~ 5 -46 de X 3- 6~ 4 +8X 2 -9+15X .
22 . 8a 4 b+a 3 b 2 - 15a 2 b 3 -45ab 4 -8 de a 5 -26a 3 b 2 +8ab 4 -b 5 +6 .
23 . 23~ 3 +8~ 4 -15~ 5 -8~-5 de ~' ~ +~ 3 +~ 2 + 9 .
24 . 7~ 7 +5~ 5 -23~ 3 +51~+36 de ~8 -~ 6 +3~ 4 -5~ 2 -9 .
25 . ~7 -60~ 4 ~3 +90~3~ 4 -50~~e-~2 ~5 de ~ 7 -3~ 5 ~2 +35~ 4 ~3 -8~ 2 ~5 +60 .
26 . a~ +2-5a~ + 1 -6a ~ de a-3-8a-1-5.
27 . Sa ~-1 +5a~ - 2 +7a~+a~ -3 de -8a~+~6a '+15a 2+ a~-3 .
28 . 31~a+ 1 - 9~ ~ + 2 -~ a + 4 -18~~ -1 de 15~~+ 3 +5~a +2- 6~a+41~a -1 .
29 . ~2a~ -2 -5a ~-~- a ~' - Sa ~ 4 de 9a ~-1 -2~a~ -2 +26a~ -3 +14a~ -5 .
30 . -~~+ 4 -6~ ~+1- 23~ ~-2 -i~ ~-1 de -15~~ 1 ;'+5O~~~+ 1 -14~~-6~~ -1 +8~~ -2 .

1 .
RESTA ~ 51
7 . a 3 ~e~~a~ -8a 2 b+6ah 2 -b 3 .
8 . ~4 ~e~~a~ -5~ 3 ~+7~ 2 ~ 2 -8~~ 3 .
9 . ~4 ~e~~a~ a 3 ~-a 4 +7a 2 ~2 -18a~ 3 +5~ 4 .
10 . 16 ~e~~a~ b-a+c+d-14 .
11 . ~2 -1 ~e~~a~ ~~+~ 2 .
12 . a 3 +6 ~e~~a~ 5a 2 b-8ab 2 +b 3 .
13 . Re~~a~ -5~-~+17~~ 2 -5 de ~ 3 +~ 3 .
14 . Re~~a~ 9~ 3 ~-15~~ 3 -8~ 2 ~ 2 de ~ 4 -1 .
15 . Re~~a~ -~ ~ a 4 b+2a 2 b 3 +8a 3 b 2 -4ab 4 de a 5 +b 5
16 . Re~~a~ 5~ 3 -25~ de ~ 4 +~ 2 +50 .
17 . Re~~a~ 9~'+17~ 4 -~ 3 +18~ 2 de ~e+~-41 .
18 . Re~~a~ -15a 5 b+17a 3 b3 -14ab 5 -be de a 8 +9a 4 b 2 +a 2 b 4 .
19 . Re~~a~ -~-+5~-34 de ~ 4 +~ 3 -11~ .
20 . Re~~a~ ~~ 2 ~~7~~ 2 -3~ 3 de ~3 -1 .
42 "STA DE POLINOMIOS CON COEFICIENTES FRACCIONARIOS
E ~e~i ~~~~
(1) De ~~3 ~e~~a~ -
1 ~3 - 2 ~~2 + 3 ~2~ - 1 ~~.
5 _ 3 4 2
8 f 3
Te~d~e~~~ : 5 ~~
Te~d~e~~~ : - a~b- - gab - 8
EJERCICIO 24
Dc :
i ~ ~ 3 ~2 ~ J 2 ~~2 ~ 2~ 3
'~3 _~-~
.. _ ~~2 . . 2 ~' . R .
(2) Re~~a~ -4a 3 b 3 -
1
1ab+ 2 a 2 b 2 -9 de -dab +~a 2 b 2 -8 .
4a 3 b 3 -- -a b 2 ~~ab - 9
4a 3 b 3 - 2a 2 b 2 -- 2ab -- 1 .
1
a- ~e~~a~ -
1
4 a-
~
-
1
.~ ab +
2
6= .
2
2 . 15 ~e~~a~ . ~~ + 3~~ - 9-
3 .
3
-bc ~e~~a~ - 3 ab + a bc -
2
-cd .
4 .
5 .
6 .
R .
1 " 4 2 1
-a--b ~e~~a~ -a+-b,,
_ ~ 5 9 2
2 X 2 - -~- ~e~~a~
5
~~ + 1 ~2 -
11 .
~~3 +
~
~3 ~e~~a~ - _~e~ + = ~~2 - 1 ~3 .
9 9 2 8 5

5 2
W .
~
1 .
2 .
7 .
8 .
9 .
10 .
ALGEBRA
EJERCICIO 25
Re~~a~ :
= a"+ '-ab -
3
-b 2 ~e~~a~
7 3 5
8 5 1 ,
-~- + ---~~ - -~e~~a~
5
a- , -~ a 2 - a + ~ ~e~~a~ -
12 . 1 7+ 3b- 7 c+ bd
1 .
3
a 2 de
3
a 2 -
~
a . 4,
2 . 3a- 3 b de Sa+6b-5 . 6 .
5
3 . 3~'~ de ~ 3 + 3~2 ~ -6 . 6 .
5 1 1
ab -
14 a 2 + 2 ~.
3 3
- ~~2 + 2~ 2 - - ~~.
7 A 7
8a~+10 +- 8 .
77,3 + ~
` ~~2 - 7 ~3 ~e~~a~ -2 1 ~2 ~ +
~
~~2 + ~3 -
~
2
3 3 5 5
11 .
~~~ + -- ~3 ~ - - ~~3 + 3 ~' ~e~~a~ ~4 + 8 ~2~2 - f ~~3 + 6 ~~.
~e~~a~ - Y3 1 b + 3 c
- - d + .
1--a- 3b+
c de a+b-c .
i~ + ~ - ~ de - i~ + 5 ~+ 1~ .
3 c -
3a 1 - -ab 2 +6 de 3a-b+ ;ab-- 3 .
7 2 2 5 1
- ~ 4 + -~ ' ~' -
1)
~~3 de -- 1 i~- 3 ~ +
1a 111-~ 2 +
~
~~3 - 6 .
~ + 3 ~ 3 ~- -
~i
~~4 ---~5 de - ~4 ~ + ~'~ 2 +
3
~2 ~3 +
~
~~4 - 7 .
7 14
~0 -
0
~4~2 + 11~'~ 4 - ~~ + ~~5 de -~5 ~ +
3
~4 ~2 -
8
~3~ 3 - ~2~ 1 + ~~ + 3)' 6 .
-( ; ~2 ~+ _~~ 2 - , ; ~3 +6 de _~~ 2 - ~~'-~+ 3~~ ; -- 3- 2
2 1 7 , 5 3 3 3 5
- -MI , + -~~ - -~ , '~~ + -?~-'~ 4 - - de -M4 ~" - -~ 2 ~' + --~ 6 .
I~ 3 20 14 - 10 9
- S c~d + 3d5 - 3 c~d2 + 3 cd 4 de 3 c , + 1 c 2 d 3 - 1 d 5 +
3
c 1 d 2 + -c Id - 35 .
11 13 G 4 9 - 3 12 22
EJERCICIO 26
Efec~~a~ ~a~ ~e~~a~ ~ig~ie~~e~ ~ ha~~a~ e~ ~a~~~ ~~~~~ic~ de~ ~e~~~~ad~
~a~a a=1, b=2, c=3, ~=4, ~=5, ~= 3 , ~==
2 5
De :
a2 -ab ~e~~a~ 3ab+b 2 .
a 3 +b- ; ~e~~a~ -5a 2 b+6ah 2 -2h 3 .
1 1 5
3 . -a ~e~~a~ -b -
3 c + a .
4 . 31~ 2 -5~ 2 ~e~~a~ ~2 +8~~+10~' .
5 . ~ -18~ 2 ~'- ~ 15~ 4 ~e~~a~ -1(i~ 33 ~-6~~ 3 +9~a .
6 . a~-7a ~~2 +~~ 3 ~e~~a~ -5a~ 2 + 8a 2 ~-5~1 3 .
7 . 3
a 2 + h
ab -
3 b 2 ~e~~a~ -a 2 + ab - 1 b 2 .
2 3 1 3 3 - I ~ 1 1
8 . .1 ~~
-~ +
4
~ ~- - -~ ~e~~a~ - ~
6
~-~ - 4 ~~- -
2
~3 .

Re~~a~ :
9 . a 4 b 2 -5a 3 b3 de a~-3a 2 b4+b~ .
10 . 15ab de -ab+~0~~-8~~ .
(2)
14 . a~-1 - 9a~ -3 + a~ -2 de
SUMA Y RESTA COMBINADAS
13 . 4 ~3 - 4 ~~2 - 3 de ~3 + ~~2 ~ - 5 ~~2 .
11 . ~~a 2 b-9ab 2 +b 3 de a 3 . '
12 .
3 6 8
~2 + ~ - de Q4~ 4.
2 5
a~-1 + a ~ - -Wa~-3 + a~-2 .
SUMA Y RESTA COMBINADAS
43 SUMA Y RESTA COMBINADAS DE POLINOMIOS
CON COEFICIENTES ENTEROS
E~e~~~~~
(1) De a 2 ~e~~a~ ~a ~~~a de 3ab - 6 ~ 3a 2 - 8ab + 5 .
3a 2 - 8ab + 5
Efec~~e~~~ ~~i~e~~ ~a ~~~a : 3ab - 6
3a 2 -5ab-1
E~~a ~~~a, ~~e e~ e~ ~~~~~ae~d~, ha~ ~~e ~e~~a~~a de a'-' ~~e
e~ e~ ~i~~e~d~, ~~eg~ deba~~ de a 2 e~c~ib~ 3a 2 - 5ab - 1
c~~ ~~~ ~ig~~~ ca~biad~~, ~ ~e~d~e~~~ : _
0 53
a 2
-3a 2 +5ab+1
-2a 2 +5 +1 . R .
De ~ 3 - 4~ 2 ~ + 5~ 3 ~e~~a~ ~a ~~~a de -~ 3 + 5~ 2 ~ - 6~~ 2 + ~ 3 c~~
-6~ 2 ~ + 9~~ 2 - 16~~ .
- ~~ + 5~ 2 ~ - 6~~ 2 + ~ 3
Efec~~e~~~ ~~i~e~~ ~a ~~~a : - 6~ 2 ~ + 9~~ 2 - 16~ 3
- ~3 - ~2 ~ + 3~~ 2 - 15~ 3 .
E~~a ~~~a, ~~e e~ e~ ~~~~~ae~d~, ~e~g~ ~~e ~e~~a~~a ~3 - 4~ 2 ~ + 5~ 3
de ~3 - 4~ 2 ~ + 5~ 3 ~~e e~ e~ ~i~~e~d~, ~~eg~ de- ~3 + ~2 ~ - 3~~ 2 + I5~ 3
ba~~ de e~~e ~i~~e~d~ e~c~ibi~~ e~ ~~~~~ae~d~ c~~
3 - 3~ 2 ~ - 3~~ 2
~~~ ~ig~~~ ca~biad~~ ~ ~e~d~e~~~ : _ 2~ + 20~ . R .
(3) De ~a ~~~a de ~ 3 +4X 2 -6 ~ - 5X
2- 1 1 ~ + 5 ~e~~a~
~3 + 42 - 6
Efec~~e~~~ ~a ~~~a : - 5~ 2 - 11 ~ + 5
X 3- X 2 -11X-1
E~~a ~~~a e~ e~ ~i~~e~d~, ~~eg~ deba~~ de e~~a e~- - 4 ~
3 - ~2 - ~ ~~ - 1
c~ibi~~ e~ ~~~~~ae~d~ ~ 4 - 1 c~~ ~~~ ~ig~~~ ca~bia-
~ + 1
d~~ ~ ~e~d~e~~~ : - - J T - ~ 4 + ~3 - ~ 2 - 11 ~ R .

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