SlideShare uma empresa Scribd logo
1 de 12
Baixar para ler offline
F. Cano Cuenca 1 Mathematics 4º ESO
Unit 2: POLYNOMIALS.
ALGEBRAIC FRACTIONS
2.1.- DIVISION OF POLYNOMIALS
Division of monomials
To divide a monomial by a monomial, divide the numerical coefficients and then
subtract the exponents of the same variables.
Examples:
5
2
3
15x
3x
5x
= −
−
3
3
6x 6
55x
=
4 6
3 2
4
12x y
4x y
3xy
=
Division of polynomials
The division of polynomials is similar to the division of natural numbers. When
you divide polynomials you get a quotient and a remainder.
Example: 4 2
A(x) 6x 8x 7x 40= + + + 2
B(x) 2x 4x 5= − +
The quotient is 2 17
Q(x) 3x 6x
2
= + + and the remainder is
5
R(x) 11x
2
= − .
( )4 2 2 2 17 5
6x 8x 7x 40 2x 4x 5 3x 6x 11x
2 2
   
+ + + = − + ⋅ + + + −   
   
4 2
6x 8x 7x 40+ + + 2
2x 4x 5− +
4 3 2
6x 12x 15x− + − 2 17
3x 6x
2
+ +
3 2
12x 24x 30x− + −
2 85
17x 34x
2
− + −
5
11x
2
−
3 2
12x 7x 7x− +
2
17x 23x 40− +
F. Cano Cuenca 2 Mathematics 4º ESO
In general, if you divide the polynomial A(x) by the polynomial B(x) and the
quotient and the remainder are Q(x) and R(x) respectively,
you can write that: A(x) B(x) Q(x) R(x)= ⋅ + .
When the remainder is 0, we have that A(x) B(x) Q(x)= ⋅ . In this case, the
polynomial A(x) is divisible by B(x), that is, B(x) is a factor or divisor of A(x).
Exercise 1:
Work out the following divisions of polynomials.
a) ( ) ( )5 4 3 2
x 7x x 8 : x 3x 1− + − − +
b) ( ) ( )5 4 3 2 2
4x 20x 18x 28x 28x 6 : x 5x 3+ − − + − + −
c) ( ) ( )4 2 2
6x 3x 2x : 3x 2+ − +
d) ( ) ( )2 3 2
45x 120x 80x : 3x 4+ + +
Division of a polynomial by x-a. Ruffini’s rule
It is very common to divide a polynomial by x a− . For example:
The quotient is 3 2
7x 10x 30x 4+ + − and the remainder is 5− .
( )( ) ( )4 3 3 2
7x 11x 94x 7 x 3 7x 10x 30x 4 5− − + = − + + − + −
A(x) B(x)
Q(x)R(x)
− − +4 3
7x 11x 94x 7 x 3−
− +4 3
7x 21x + + −3 2
7x 10x 30x 4
− +3 2
10x 30x
− +2
30x 90x
− +4x 7
3
10x
−2
30x 94x
−4x 12
−5
F. Cano Cuenca 3 Mathematics 4º ESO
But this division can also be done using Ruffini’s rule:
We start writing the coefficients of the dividend and the number a.
QUOTIENT: 7 10 30 4− , that is, 3 2
7x 10x 30x 4+ + −
REMAINDER: 5−
Notice that Ruffini’s rule’s steps are exactly the same as the steps of the long
division. The advantage of Ruffini’s rule is that you only work with the
coefficients and only do the essential operations.
IMPORTANT!!
Exercise 2:
Use Ruffini’s rule for doing the following divisions of polynomials.
a) ( ) ( )4 2
5x 6x 11x 13 : x 2+ − + −
b) ( ) ( )5 4
6x 3x 2x : x 1− + +
c) ( ) ( )4 3 2
3x 5x 7x 2x 13 : x 4− + − + −
d) ( ) ( )4 3 2
6x 4x 51x 3x 9 : x 3+ − − − +
QUOTIENT ’S COEFFICIENTS REMAINDER
Ruffini’s rule only works when you divide a polynomial by a linear factor x a− .
F. Cano Cuenca 4 Mathematics 4º ESO
2.2.- RUFFINI’S RULE’S USES
Look at the division ( ) ( )3 2
2x 8x 31x 42 : x 6− − + −
2 -8 -31 42
6 12 24 ( 7) 6− ⋅
2 4 -7 0
The quotient is 2
2x 4x 7+ − and the remainder is 0.
Therefore, you can write that ( )( )3 2 2
2x 8x 31x 42 x 6 2x 4x 7− − + = − + − .
Then, ( )x 6− is a factor of the polynomial 3 2
2x 8x 31x 42− − + , that is,
the polynomial 3 2
2x 8x 31x 42− − + is divisible by ( )x 6− .
Notice that 6 is a divisor of 42.
So if you are looking for factors of a polynomial P(x), have a try with the linear
factors (x a)− where “a” is a divisor of the constant term of P(x).
Exercise 3:
Find two linear factors of the polynomial 4 3 2
x 3x 2x 10x 12+ − − − .
Exercise 4:
Check if the following polynomials are divisible by x 3− or x 1+ .
a) 3 2
A(x) x 3x x 3= − + −
b) 3 2
B(x) x 4x 11x 30= + − −
c) 4 3 2
C(x) x 7x 5x 13= − + −
The Remainder Theorem
Remember that you can calculate the number value of a polynomial at a given
value of the variable.
When the coefficients of a polynomial P(x) are integers, if(x a)− is a
factor of P(x) and “a” is also an integer number, then “a” is a divisor of the
constant term of P(x).
F. Cano Cuenca 5 Mathematics 4º ESO
Example: Calculate the number value of 3 2
P(x) 2x x 4x 2= − − + at x 3= −
( ) ( ) ( ) ( )
3 2
P( ) 2 4 2 2 27 9 123 3 3 2 54 9 12 2 493= − − + = ⋅ − − + + = − − + +− − = −− −
The Remainder Theorem states:
Proof:
If x a P(a) (a a) Q(a) R R= ⇒ = − ⋅ + = P(a) R⇒ =
Exercise 5:
Use Ruffini’s rule to calculate P(a) in the following cases.
a) 4 2
P(x) 7x 5x 2x 24= − + − , a 5= − , a 10=
b) 3 2
P(x) 3x 8x 3x= − + , a 1= , a 8=
Exercise 6:
Find the value of m so that the polynomial 3 2
P(x) x mx 5x 2= − + − is divisible by
x 1+ .
Exercise 7:
The remainder of the division ( ) ( )4 3
2x kx 7x 6 : x 2+ − + − is 8− . What is the
value of k?
2.3.- FACTORIZING POLYNOMIALS
Roots of a polynomial
A number “a” is called a root of a polynomial P(x) if P(a) 0= . The roots (or
zeroes) of a polynomial are the solutions of the equation P(x) 0= .
Examples: a) The numbers 1 and 1− are roots of the polynomial 2
P(x) x 1= − .
2
P(1) 1 1 0= − = 2
P( 1) ( 1) 1 0− = − − =
b) Find the roots of the polynomial 2
P(x) x 5x 6= − + .
The number value of the polynomial P(x) at x a= is the same as the
remainder of the division P(x) : (x a)− . That is, P(a) R= .
P(x) (x a) Q(x) R= − ⋅ +
F. Cano Cuenca 6 Mathematics 4º ESO
The roots of P(x) are the solutions of the equation P(x) 0= .
2 1
2
x 2
P(x) 0 x 5x 6 0
x 3
 =
= ⇒ − + = ⇒ 
=
The roots of the polynomial 2
P(x) x 5x 6= − + are 2 and 3.
One of the most important uses of Ruffini’s rule is to find the roots of a
polynomial.
Remember that the remainder of the division P(x) : (x a)− is the same as P(a).
Therefore, if the remainder is 0, then P(a) 0= , so the number “a” is a root of
P(x).
Examples: Find a root of the polynomial 4 3 2
P(x) x 2x 7x 8x 12= + − − +
1 2 -7 -8 12
1 1 3 -4 -12
1 3 -4 -12 0
The remainder of the division P(x) : (x 1)− is 0. Then, 1 is a root of P(x).
An interesting theorem about polynomials is the Fundamental Theorem of
Algebra:
Let n n 1 n 2 2
n n 1 n 2 2 1 0
P(x) a x a x a x ... a x a x a− −
− −
= + + + + + + be any polynomial of degree
n. If 1 2 3 n
r, r , r ,...,r are its n roots, then the polynomial can be factorized as:
Example: Find the roots of the polynomial 2
P(x) 2x x 1= − − and factorize it.
1
2
2
x 1
P(x) 0 2x x 1 0 1
x
2
 =

= ⇒ − − = ⇒ 
= −

If the remainder of the division P(x) : (x a)− is 0,
then the number “a” is a root of P(x).
Any polynomial of degree n has n roots.
(These roots can be real or complex numbers).
n 1 2 n 1 n
P(x) a (x r) (x r ) .... (x r ) (x r )−
= − ⋅ − ⋅ ⋅ − ⋅ −
F. Cano Cuenca 7 Mathematics 4º ESO
The roots of 2
P(x) 2x x 1= − − are 1 and
1
2
− . The factorization of P(x) is
( ) ( )1 1
P(x) 2 x 1 x 2 x 1 x
2 2
    
= − − − = − +    
    
.
Factorization techniques
You have seen before the factorization of a polynomial in the previous example.
Factorize a polynomial means to write it as a product of lower degree
polynomials.
Examples: ( )( )2
x 9 x 3 x 3− = + − ( )( )3 2
x 5x 6x x x 2 x 3− + = − −
There are different techniques for factorizing polynomials:
1) Taking out common factor.
Example: ( )2
8x 2x 2x 4x 1− = −
2) Using polynomial identities (the square of the sum, the square of the
difference, the product of a sum and a difference).
Examples: ( ) ( )( )
22
x 10x 25 x 5 x 5 x 5+ + = + = + +
( ) ( )( )
22
x 6x 9 x 3 x 3 x 3− + = − = − −
( )( )2
x 16 x 4 x 4− = + −
3) Using the Fundamental Theorem of Algebra.
Example: The roots of the polynomial 2
P(x) x x 6= − − + are 2 and 3− .
( )( )( )2
P(x) x x 6 1 x 2 x 3= − − + = − − +
Exercise 8:
Take out common factor or use the polynomial identities to factorize the
following polynomials.
a) 2
3x 12x− b) 3 2
4x 24x 36x− + c) 2 4
45x 5x−
d) 4 2 3
x x 2x+ + e) 6 2
x 16x− f) 4
16x 9−
Exercise 9:
Use the Fundamental Theorem of Algebra to factorize the following polynomials.
a) 2
x 4x 5+ − b) 2
x 8x 15+ +
c) 2
7x 21x 280− − d) 2
3x 9x 210+ −
F. Cano Cuenca 8 Mathematics 4º ESO
4) Using Ruffini’s rule.
Example: 3 2
P(x) x 2x 5x 6= − − +
( )( ) ( )( )( )3 2 2
P(x) x 2x 5x 6 x 1 x x 6 x 1 x 2 x 3= − − + = − − − = − + −
5) A combination of the previous ones.
Examples:
a) ( ) ( )
25 4 3 3 2 3
P(x) 12x 36x 27x 3x 4x 12x 9 3x 2x 3= − + = − + = −
If we solve the equation P(x) 0= , we get the roots of P(x).
( ) ( ) ( )
23
P(x) 0 3x 2x 3 0 3 x x x 2x 3 2x 3 0= ⇒ − = ⇒ ⋅ ⋅ ⋅ ⋅ − ⋅ − = ⇒
1 2 3 4 5
3 3
x 0, x 0, x 0, x , x
2 2
⇒ = = = = =
The polynomial has a double root and a triple root.
b) 3
P(x) x x 6= − −
Firstly, we use Ruffini’s rule:
Secondly, we find the roots of the polynomial 2
x 2x 3− + .
1 -2 -5 6
1 1 -1 -6
1 -1 -6 0
-2 -2 6
1 -3 0
Taking out
common factor
Using polynomial
identities
1 0 -1 6
-2 -2 4 -6
1 -2 3 0
F. Cano Cuenca 9 Mathematics 4º ESO
2
x 2x 3 0− + = a 1, b 2, c 3= = − =
( )
2
2 2 4 1 3 2 8
x
2 1 2
± − − ⋅ ⋅ ± −
= =
⋅
The roots of the polynomial 2
x 2x 3− + are not real numbers.
So the factorization of P(x) is ( )( )2
P(x) x 2 x 2x 3= + − + .
Exercise 10:
Factorize the following polynomials.
a) 2
3x 2x 8+ − b) 5
3x 48x− c) 3 2
2x x 5x 12+ − +
d) 3 2
x 7x 8x 16− + + e) 4 3 2
x 2x 23x 60x+ − − f) 4 3 2
9x 36x 26x 4x 3− + + −
Exercise 11:
Take out common factor in the following expressions.
a) 3x(x 3) (x 1)(x 3)− − + −
b) (x 5)(2x 1) (x 5)(2x 1)+ − + − −
c) (3 y)(a b) (a b)(3 y)− + − − −
Exercise 12:
Write second degree polynomials whose roots are:
a) 7 and -7 b) 0 and 5 c) -2 and -3 d) 4 (double)
Exercise 13:
Prove that the polynomial n
x 1− is divisible by x 1− for any value of n. Find the
general expression for the quotient of this division.
Exercise 14:
The remainder of the division 2
P(x) : (x 1)− is 4x 4+ . Find the remainder of the
division P(x) : (x 1)− .
Exercise 15:
Prove that the polynomial 2
x (a b)x ab+ + + is divisible by x a+ and by x b+ for
any values of a and b. Find its factorization.
2 8i 2 2 2i2 8
1 2i
2 2 2
+ ++ −
= = = + ∉ »
2 8i 2 2 2i2 8
1 2i
2 2 2
− −− −
= = = − ∉ »
F. Cano Cuenca 10 Mathematics 4º ESO
2.4.- ALGEBRAIC FRACTIONS
An algebraic fraction is the quotient of two polynomials, that is,
P(x)
Q(x)
Examples: 2
2x 4
x 5x 3
+
− +
1
x 5−
4
3 2
x 5x 1
x x
− +
+
2x 5
11
+
The same calculations that you do with numerical fractions can be done with
algebraic fractions.
Simplification of algebraic fractions
To simplify algebraic fractions:
• Factorize the polynomials.
• Cancel out the common factors.
Example:
( )2
2
x 1x 1
x 2x 3
+−
=
− −
( )
( )
x 1
x 1
−
+ ( )
x 1
x 3x 3
−
=
−−
Addition, subtraction, product and division of algebraic fractions
You can add, subtract, multiply an divide algebraic fractions in the same way
that you do in simple arithmetic.
Examples:
a)
( )
( )( ) ( ) ( )
( ) ( )
2 22
x 7 x 2 2x 1
x x 1
x 1 x 7 x 2 x 2x 1 2x 7x 9
x x 1x 1 x x 1x
+ + −+ − −
− +
− + + + +
=
+
=
+ + +
b)
( )( )
( )
2 2
3 2 3 222
x 2 x 1 xx x 2x 2 x x 2
2x x 2
2 x 1
2x 1 xx 2 1x xx
+ − − + −+ − + −
= = =
+ ++
⋅
+
c)
( )( )
( )( )
( ) ( )
2
2
x 5 x x 2x 5 x 2x
x
x 5 2x 1
:
x 2 x 2x 12x 2
+ ++ +
= =
++ +
+ −
+ ( )x 2+ ( )
( ) 2x 5 x x 5x
2x 1 2x 12x 1
+ +
= =
+ ++
d)
( )( )
( )
( )( )( )
( ) ( )
2 2
2 2 2 2
x 9 xx 9 x 4 x 9 x 4 x 2x 3
:
x 2x 2
4 x 3
:
x x x2 x x 2 x2 x 3
− − − − −−
= = =
−+− +
− −
⋅
−
−
+
( ) ( )x 3 x 3+ −
=
( )x 2+ ( )( )
( )
x 2 x 2
x 2
− −
+ ( )x x 3−
3 2
x x 8x 12
x
− − +
=
( )LCM x, x(x 1), (x 1) x(x 1)− − = −
F. Cano Cuenca 11 Mathematics 4º ESO
Exercise 16:
Simplify the following algebraic fractions.
a)
2
3
2x 6x
4x 2x
−
−
b)
( ) ( )
( ) ( )
2
2
x 3 x x 3
x 3 x x 2
− +
− +
c)
3 2
3 2
x 3x x 3
x 3x
+ + +
+
d)
3 2
3 2
x 5x 6x
x x 14x 24
− +
− − +
Exercise 17:
Work out and simplify.
a)
2
2
2x 1 x 5
x 3 x 3x
+ +
−
+ +
b)
2
2
3 x x
x x 1 x 1
 
− 
+ − 
c)
2
5x 10 x 9
x 3 x 2
− −
⋅
+ −
d) 2
3x 1 x 3 2x 5
x x 2x 2x
− + +
− +
−−
e)
2
2x 1 x
:
2x 1 4x 2
+
− −
f)
2
x 1 1
:
x 1 x x 1
 
− 
− − 
Exercise 18:
Translate into algebraic language. (You can only use one unknown).
a) The quotient between two even consecutive numbers.
b) A number minus its inverse.
c) The inverse of a number plus the inverse of twice that number.
d) The sum of the inverses of two consecutive numbers.
Exercise 19:
Use polynomials to express the area and the volume of this prism.
Exercise 20:
A tap takes x minutes to fill a tank. Another tap takes 3 minutes less than the
first one to fill the same tank. Express as a function of x the part of the tank to
be filled if we turn both taps on for one minute.
F. Cano Cuenca 12 Mathematics 4º ESO
Exercise 21:
We mix “x” kg of a paint that costs 5 €/kg with “y” kg of another paint that
costs 3 €/k. Express the price of 1 kg of the mixture as a function of “x” and
“y”.
Exercise 22:
Two cyclists start at the same time from opposite ends of a course that is 60
miles long. One cyclist is riding at x mph and the second cyclist is riding at x 3+
mph. How long after they begin will they meet? (Give the answer as a function of
x).
Exercise 23:
A rhombus is inscribed in a rectangle whose sides are x
an y. Express the perimeter of the rhombus as a
function of x and y.
Exercise 24:
The sides AB and BC in the rectangle ABCD are 3 and 5 cm respectively. If
AA' BB' CC' DD' x= = = = , express the area of the rhomboid A’B’C’D’ as a
function of x.
Exercise 25:
The number value of the polynomial 34
x
3
π express the volume of a sphere with
radius x, and the number value of the polynomial 2
4 xπ express the area of a
sphere with radius x.
a) Is there any sphere whose volume (expressed in m3
) is the same as its
area (expressed in m2
)? If your answer is affirmative, find the radius of
that sphere.
b) Find the relation between the area of a sphere and the area of a maximum
circle in that sphere.
c) Use a polynomial of x to express the volume of the cylinder circumscribed
in a sphere with radius x. Find the relation between this polynomial and
the one which express the volume of the sphere.

Mais conteúdo relacionado

Mais procurados

Exponential and logarithmic functions
Exponential and logarithmic functionsExponential and logarithmic functions
Exponential and logarithmic functionsNjabulo Nkabinde
 
Trigonometric Functions and their Graphs
Trigonometric Functions and their GraphsTrigonometric Functions and their Graphs
Trigonometric Functions and their GraphsMohammed Ahmed
 
Linear Equations Slide Share Version Exploded[1]
Linear  Equations Slide Share Version Exploded[1]Linear  Equations Slide Share Version Exploded[1]
Linear Equations Slide Share Version Exploded[1]keithpeter
 
Class 10 Maths Ch Polynomial PPT
Class 10 Maths Ch Polynomial PPTClass 10 Maths Ch Polynomial PPT
Class 10 Maths Ch Polynomial PPTSanjayraj Balasara
 
Applied Calculus Chapter 4 multiple integrals
Applied Calculus Chapter  4 multiple integralsApplied Calculus Chapter  4 multiple integrals
Applied Calculus Chapter 4 multiple integralsJ C
 
Presentaton on Polynomials....Class 10
Presentaton on Polynomials....Class 10 Presentaton on Polynomials....Class 10
Presentaton on Polynomials....Class 10 Bindu Cm
 
4.1 Defining and visualizing binary relations
4.1 Defining and visualizing binary relations4.1 Defining and visualizing binary relations
4.1 Defining and visualizing binary relationsJan Plaza
 
Indirect variation notes
Indirect variation notesIndirect variation notes
Indirect variation noteskke18914
 
Section 6.3 properties of the trigonometric functions
Section 6.3 properties of the trigonometric functionsSection 6.3 properties of the trigonometric functions
Section 6.3 properties of the trigonometric functionsWong Hsiung
 
Rational expressions
Rational expressionsRational expressions
Rational expressionsMark Ryder
 
Section 5.4 logarithmic functions
Section 5.4 logarithmic functions Section 5.4 logarithmic functions
Section 5.4 logarithmic functions Wong Hsiung
 
A25-7 Quadratic Inequalities
A25-7 Quadratic InequalitiesA25-7 Quadratic Inequalities
A25-7 Quadratic Inequalitiesvhiggins1
 
7 cavalieri principle-x
7 cavalieri principle-x7 cavalieri principle-x
7 cavalieri principle-xmath266
 
1.3 solving equations t
1.3 solving equations t1.3 solving equations t
1.3 solving equations tmath260
 
Addition and subtraction of rational expression
Addition and subtraction of rational expressionAddition and subtraction of rational expression
Addition and subtraction of rational expressionMartinGeraldine
 
M8 acc lesson 1 1 polynomials
M8 acc lesson 1 1 polynomialsM8 acc lesson 1 1 polynomials
M8 acc lesson 1 1 polynomialslothomas
 

Mais procurados (20)

Polynomials
PolynomialsPolynomials
Polynomials
 
Exponential and logarithmic functions
Exponential and logarithmic functionsExponential and logarithmic functions
Exponential and logarithmic functions
 
Practica 18 división de polinomios ii solucion
Practica 18  división de polinomios ii solucionPractica 18  división de polinomios ii solucion
Practica 18 división de polinomios ii solucion
 
Trigonometric Functions and their Graphs
Trigonometric Functions and their GraphsTrigonometric Functions and their Graphs
Trigonometric Functions and their Graphs
 
Linear Equations Slide Share Version Exploded[1]
Linear  Equations Slide Share Version Exploded[1]Linear  Equations Slide Share Version Exploded[1]
Linear Equations Slide Share Version Exploded[1]
 
Class 10 Maths Ch Polynomial PPT
Class 10 Maths Ch Polynomial PPTClass 10 Maths Ch Polynomial PPT
Class 10 Maths Ch Polynomial PPT
 
Applied Calculus Chapter 4 multiple integrals
Applied Calculus Chapter  4 multiple integralsApplied Calculus Chapter  4 multiple integrals
Applied Calculus Chapter 4 multiple integrals
 
Presentaton on Polynomials....Class 10
Presentaton on Polynomials....Class 10 Presentaton on Polynomials....Class 10
Presentaton on Polynomials....Class 10
 
Vector Space.pptx
Vector Space.pptxVector Space.pptx
Vector Space.pptx
 
4.1 Defining and visualizing binary relations
4.1 Defining and visualizing binary relations4.1 Defining and visualizing binary relations
4.1 Defining and visualizing binary relations
 
Indirect variation notes
Indirect variation notesIndirect variation notes
Indirect variation notes
 
Factor by grouping
Factor by groupingFactor by grouping
Factor by grouping
 
Section 6.3 properties of the trigonometric functions
Section 6.3 properties of the trigonometric functionsSection 6.3 properties of the trigonometric functions
Section 6.3 properties of the trigonometric functions
 
Rational expressions
Rational expressionsRational expressions
Rational expressions
 
Section 5.4 logarithmic functions
Section 5.4 logarithmic functions Section 5.4 logarithmic functions
Section 5.4 logarithmic functions
 
A25-7 Quadratic Inequalities
A25-7 Quadratic InequalitiesA25-7 Quadratic Inequalities
A25-7 Quadratic Inequalities
 
7 cavalieri principle-x
7 cavalieri principle-x7 cavalieri principle-x
7 cavalieri principle-x
 
1.3 solving equations t
1.3 solving equations t1.3 solving equations t
1.3 solving equations t
 
Addition and subtraction of rational expression
Addition and subtraction of rational expressionAddition and subtraction of rational expression
Addition and subtraction of rational expression
 
M8 acc lesson 1 1 polynomials
M8 acc lesson 1 1 polynomialsM8 acc lesson 1 1 polynomials
M8 acc lesson 1 1 polynomials
 

Destaque

Edexcel Maths – Core 2 – Algebraic Division and Remainder Theorem
Edexcel Maths – Core 2 – Algebraic Division and Remainder TheoremEdexcel Maths – Core 2 – Algebraic Division and Remainder Theorem
Edexcel Maths – Core 2 – Algebraic Division and Remainder TheoremUmayr Dawood
 
A19 1 square roots
A19 1 square rootsA19 1 square roots
A19 1 square rootsvhiggins1
 
Roots of a quadratic equation1
Roots of a quadratic equation1Roots of a quadratic equation1
Roots of a quadratic equation1Wilson ak
 
Algebraic fractions 4
Algebraic fractions 4Algebraic fractions 4
Algebraic fractions 4shaminakhan
 
Oct.19 Substitution And Properties Of Roots
Oct.19 Substitution And Properties Of RootsOct.19 Substitution And Properties Of Roots
Oct.19 Substitution And Properties Of RootsRyanWatt
 
Solving quadratic equations
Solving quadratic equationsSolving quadratic equations
Solving quadratic equationssrobbins4
 
Quadratic And Roots
Quadratic And RootsQuadratic And Roots
Quadratic And RootsPeking
 
The remainder theorem powerpoint
The remainder theorem powerpointThe remainder theorem powerpoint
The remainder theorem powerpointJuwileene Soriano
 
Factor Theorem and Remainder Theorem
Factor Theorem and Remainder TheoremFactor Theorem and Remainder Theorem
Factor Theorem and Remainder TheoremRonalie Mejos
 
Algebraic fractionns 2
Algebraic fractionns 2Algebraic fractionns 2
Algebraic fractionns 2gheovani
 
Remainder and Factor Theorem
Remainder and Factor TheoremRemainder and Factor Theorem
Remainder and Factor TheoremTrish Hammond
 

Destaque (16)

Edexcel Maths – Core 2 – Algebraic Division and Remainder Theorem
Edexcel Maths – Core 2 – Algebraic Division and Remainder TheoremEdexcel Maths – Core 2 – Algebraic Division and Remainder Theorem
Edexcel Maths – Core 2 – Algebraic Division and Remainder Theorem
 
A19 1 square roots
A19 1 square rootsA19 1 square roots
A19 1 square roots
 
Roots of a quadratic equation1
Roots of a quadratic equation1Roots of a quadratic equation1
Roots of a quadratic equation1
 
Algebraic fractions 4
Algebraic fractions 4Algebraic fractions 4
Algebraic fractions 4
 
Oct.19 Substitution And Properties Of Roots
Oct.19 Substitution And Properties Of RootsOct.19 Substitution And Properties Of Roots
Oct.19 Substitution And Properties Of Roots
 
Algebraic fractions section 1
Algebraic fractions section 1Algebraic fractions section 1
Algebraic fractions section 1
 
Simplifying & solving algebraic fractions
Simplifying & solving algebraic fractionsSimplifying & solving algebraic fractions
Simplifying & solving algebraic fractions
 
Division of algebraic expressions
Division of algebraic expressionsDivision of algebraic expressions
Division of algebraic expressions
 
Solving quadratic equations
Solving quadratic equationsSolving quadratic equations
Solving quadratic equations
 
Quadratic And Roots
Quadratic And RootsQuadratic And Roots
Quadratic And Roots
 
The remainder theorem powerpoint
The remainder theorem powerpointThe remainder theorem powerpoint
The remainder theorem powerpoint
 
10.5
10.510.5
10.5
 
Factor Theorem and Remainder Theorem
Factor Theorem and Remainder TheoremFactor Theorem and Remainder Theorem
Factor Theorem and Remainder Theorem
 
Algebraic fractionns 2
Algebraic fractionns 2Algebraic fractionns 2
Algebraic fractionns 2
 
Division of polynomials
Division of polynomialsDivision of polynomials
Division of polynomials
 
Remainder and Factor Theorem
Remainder and Factor TheoremRemainder and Factor Theorem
Remainder and Factor Theorem
 

Semelhante a Unit2.polynomials.algebraicfractions

Module 1 polynomial functions
Module 1   polynomial functionsModule 1   polynomial functions
Module 1 polynomial functionsdionesioable
 
Dividing polynomials
Dividing polynomialsDividing polynomials
Dividing polynomialsEducación
 
Higher Maths 2.1.1 - Polynomials
Higher Maths 2.1.1 - PolynomialsHigher Maths 2.1.1 - Polynomials
Higher Maths 2.1.1 - Polynomialstimschmitz
 
Delos-Santos-Analyn-M.-_Repoter-No.-1-Multiplication-and-Division-of-Polynomi...
Delos-Santos-Analyn-M.-_Repoter-No.-1-Multiplication-and-Division-of-Polynomi...Delos-Santos-Analyn-M.-_Repoter-No.-1-Multiplication-and-Division-of-Polynomi...
Delos-Santos-Analyn-M.-_Repoter-No.-1-Multiplication-and-Division-of-Polynomi...polanesgumiran
 
3.2 properties of division and roots t
3.2 properties of division and roots t3.2 properties of division and roots t
3.2 properties of division and roots tmath260
 
Remainder theorem
Remainder theoremRemainder theorem
Remainder theoremRhodaLuis
 
Form 4 Add Maths Note
Form 4 Add Maths NoteForm 4 Add Maths Note
Form 4 Add Maths NoteChek Wei Tan
 
Form 4-add-maths-note
Form 4-add-maths-noteForm 4-add-maths-note
Form 4-add-maths-notejacey tan
 
Module 2 polynomial functions
Module 2   polynomial functionsModule 2   polynomial functions
Module 2 polynomial functionsdionesioable
 
Chapter 6 taylor and maclaurin series
Chapter 6 taylor and maclaurin seriesChapter 6 taylor and maclaurin series
Chapter 6 taylor and maclaurin seriesIrfaan Bahadoor
 
Polynomial- Maths project
Polynomial- Maths projectPolynomial- Maths project
Polynomial- Maths projectRITURAJ DAS
 
College algebra real mathematics real people 7th edition larson solutions manual
College algebra real mathematics real people 7th edition larson solutions manualCollege algebra real mathematics real people 7th edition larson solutions manual
College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
 
Zeros of a polynomial function
Zeros of a polynomial functionZeros of a polynomial function
Zeros of a polynomial functionMartinGeraldine
 

Semelhante a Unit2.polynomials.algebraicfractions (20)

Module 1 polynomial functions
Module 1   polynomial functionsModule 1   polynomial functions
Module 1 polynomial functions
 
Dividing polynomials
Dividing polynomialsDividing polynomials
Dividing polynomials
 
Factor theorem
Factor theoremFactor theorem
Factor theorem
 
Higher Maths 2.1.1 - Polynomials
Higher Maths 2.1.1 - PolynomialsHigher Maths 2.1.1 - Polynomials
Higher Maths 2.1.1 - Polynomials
 
Delos-Santos-Analyn-M.-_Repoter-No.-1-Multiplication-and-Division-of-Polynomi...
Delos-Santos-Analyn-M.-_Repoter-No.-1-Multiplication-and-Division-of-Polynomi...Delos-Santos-Analyn-M.-_Repoter-No.-1-Multiplication-and-Division-of-Polynomi...
Delos-Santos-Analyn-M.-_Repoter-No.-1-Multiplication-and-Division-of-Polynomi...
 
3.2 properties of division and roots t
3.2 properties of division and roots t3.2 properties of division and roots t
3.2 properties of division and roots t
 
Algebra
AlgebraAlgebra
Algebra
 
Remainder theorem
Remainder theoremRemainder theorem
Remainder theorem
 
Form 4 Add Maths Note
Form 4 Add Maths NoteForm 4 Add Maths Note
Form 4 Add Maths Note
 
Form 4-add-maths-note
Form 4-add-maths-noteForm 4-add-maths-note
Form 4-add-maths-note
 
Module 2 polynomial functions
Module 2   polynomial functionsModule 2   polynomial functions
Module 2 polynomial functions
 
Chapter 6 taylor and maclaurin series
Chapter 6 taylor and maclaurin seriesChapter 6 taylor and maclaurin series
Chapter 6 taylor and maclaurin series
 
Form 4 add maths note
Form 4 add maths noteForm 4 add maths note
Form 4 add maths note
 
Bonus math project
Bonus math projectBonus math project
Bonus math project
 
Polynomials
PolynomialsPolynomials
Polynomials
 
Quadraticequation
QuadraticequationQuadraticequation
Quadraticequation
 
Polynomial- Maths project
Polynomial- Maths projectPolynomial- Maths project
Polynomial- Maths project
 
College algebra real mathematics real people 7th edition larson solutions manual
College algebra real mathematics real people 7th edition larson solutions manualCollege algebra real mathematics real people 7th edition larson solutions manual
College algebra real mathematics real people 7th edition larson solutions manual
 
2 5 zeros of poly fn
2 5 zeros of poly fn2 5 zeros of poly fn
2 5 zeros of poly fn
 
Zeros of a polynomial function
Zeros of a polynomial functionZeros of a polynomial function
Zeros of a polynomial function
 

Último

TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...
TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...
TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...Nguyen Thanh Tu Collection
 
How to Manage Global Discount in Odoo 17 POS
How to Manage Global Discount in Odoo 17 POSHow to Manage Global Discount in Odoo 17 POS
How to Manage Global Discount in Odoo 17 POSCeline George
 
REMIFENTANIL: An Ultra short acting opioid.pptx
REMIFENTANIL: An Ultra short acting opioid.pptxREMIFENTANIL: An Ultra short acting opioid.pptx
REMIFENTANIL: An Ultra short acting opioid.pptxDr. Ravikiran H M Gowda
 
How to Add New Custom Addons Path in Odoo 17
How to Add New Custom Addons Path in Odoo 17How to Add New Custom Addons Path in Odoo 17
How to Add New Custom Addons Path in Odoo 17Celine George
 
On_Translating_a_Tamil_Poem_by_A_K_Ramanujan.pptx
On_Translating_a_Tamil_Poem_by_A_K_Ramanujan.pptxOn_Translating_a_Tamil_Poem_by_A_K_Ramanujan.pptx
On_Translating_a_Tamil_Poem_by_A_K_Ramanujan.pptxPooja Bhuva
 
Basic Civil Engineering first year Notes- Chapter 4 Building.pptx
Basic Civil Engineering first year Notes- Chapter 4 Building.pptxBasic Civil Engineering first year Notes- Chapter 4 Building.pptx
Basic Civil Engineering first year Notes- Chapter 4 Building.pptxDenish Jangid
 
Accessible Digital Futures project (20/03/2024)
Accessible Digital Futures project (20/03/2024)Accessible Digital Futures project (20/03/2024)
Accessible Digital Futures project (20/03/2024)Jisc
 
Google Gemini An AI Revolution in Education.pptx
Google Gemini An AI Revolution in Education.pptxGoogle Gemini An AI Revolution in Education.pptx
Google Gemini An AI Revolution in Education.pptxDr. Sarita Anand
 
How to setup Pycharm environment for Odoo 17.pptx
How to setup Pycharm environment for Odoo 17.pptxHow to setup Pycharm environment for Odoo 17.pptx
How to setup Pycharm environment for Odoo 17.pptxCeline George
 
Holdier Curriculum Vitae (April 2024).pdf
Holdier Curriculum Vitae (April 2024).pdfHoldier Curriculum Vitae (April 2024).pdf
Holdier Curriculum Vitae (April 2024).pdfagholdier
 
Exploring_the_Narrative_Style_of_Amitav_Ghoshs_Gun_Island.pptx
Exploring_the_Narrative_Style_of_Amitav_Ghoshs_Gun_Island.pptxExploring_the_Narrative_Style_of_Amitav_Ghoshs_Gun_Island.pptx
Exploring_the_Narrative_Style_of_Amitav_Ghoshs_Gun_Island.pptxPooja Bhuva
 
HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptx
HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptxHMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptx
HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptxmarlenawright1
 
COMMUNICATING NEGATIVE NEWS - APPROACHES .pptx
COMMUNICATING NEGATIVE NEWS - APPROACHES .pptxCOMMUNICATING NEGATIVE NEWS - APPROACHES .pptx
COMMUNICATING NEGATIVE NEWS - APPROACHES .pptxannathomasp01
 
Sociology 101 Demonstration of Learning Exhibit
Sociology 101 Demonstration of Learning ExhibitSociology 101 Demonstration of Learning Exhibit
Sociology 101 Demonstration of Learning Exhibitjbellavia9
 
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...ZurliaSoop
 
Key note speaker Neum_Admir Softic_ENG.pdf
Key note speaker Neum_Admir Softic_ENG.pdfKey note speaker Neum_Admir Softic_ENG.pdf
Key note speaker Neum_Admir Softic_ENG.pdfAdmir Softic
 
Micro-Scholarship, What it is, How can it help me.pdf
Micro-Scholarship, What it is, How can it help me.pdfMicro-Scholarship, What it is, How can it help me.pdf
Micro-Scholarship, What it is, How can it help me.pdfPoh-Sun Goh
 
NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...
NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...
NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...Amil baba
 
Unit 3 Emotional Intelligence and Spiritual Intelligence.pdf
Unit 3 Emotional Intelligence and Spiritual Intelligence.pdfUnit 3 Emotional Intelligence and Spiritual Intelligence.pdf
Unit 3 Emotional Intelligence and Spiritual Intelligence.pdfDr Vijay Vishwakarma
 

Último (20)

TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...
TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...
TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...
 
How to Manage Global Discount in Odoo 17 POS
How to Manage Global Discount in Odoo 17 POSHow to Manage Global Discount in Odoo 17 POS
How to Manage Global Discount in Odoo 17 POS
 
REMIFENTANIL: An Ultra short acting opioid.pptx
REMIFENTANIL: An Ultra short acting opioid.pptxREMIFENTANIL: An Ultra short acting opioid.pptx
REMIFENTANIL: An Ultra short acting opioid.pptx
 
How to Add New Custom Addons Path in Odoo 17
How to Add New Custom Addons Path in Odoo 17How to Add New Custom Addons Path in Odoo 17
How to Add New Custom Addons Path in Odoo 17
 
On_Translating_a_Tamil_Poem_by_A_K_Ramanujan.pptx
On_Translating_a_Tamil_Poem_by_A_K_Ramanujan.pptxOn_Translating_a_Tamil_Poem_by_A_K_Ramanujan.pptx
On_Translating_a_Tamil_Poem_by_A_K_Ramanujan.pptx
 
Basic Civil Engineering first year Notes- Chapter 4 Building.pptx
Basic Civil Engineering first year Notes- Chapter 4 Building.pptxBasic Civil Engineering first year Notes- Chapter 4 Building.pptx
Basic Civil Engineering first year Notes- Chapter 4 Building.pptx
 
Accessible Digital Futures project (20/03/2024)
Accessible Digital Futures project (20/03/2024)Accessible Digital Futures project (20/03/2024)
Accessible Digital Futures project (20/03/2024)
 
Google Gemini An AI Revolution in Education.pptx
Google Gemini An AI Revolution in Education.pptxGoogle Gemini An AI Revolution in Education.pptx
Google Gemini An AI Revolution in Education.pptx
 
How to setup Pycharm environment for Odoo 17.pptx
How to setup Pycharm environment for Odoo 17.pptxHow to setup Pycharm environment for Odoo 17.pptx
How to setup Pycharm environment for Odoo 17.pptx
 
Holdier Curriculum Vitae (April 2024).pdf
Holdier Curriculum Vitae (April 2024).pdfHoldier Curriculum Vitae (April 2024).pdf
Holdier Curriculum Vitae (April 2024).pdf
 
Exploring_the_Narrative_Style_of_Amitav_Ghoshs_Gun_Island.pptx
Exploring_the_Narrative_Style_of_Amitav_Ghoshs_Gun_Island.pptxExploring_the_Narrative_Style_of_Amitav_Ghoshs_Gun_Island.pptx
Exploring_the_Narrative_Style_of_Amitav_Ghoshs_Gun_Island.pptx
 
HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptx
HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptxHMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptx
HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptx
 
COMMUNICATING NEGATIVE NEWS - APPROACHES .pptx
COMMUNICATING NEGATIVE NEWS - APPROACHES .pptxCOMMUNICATING NEGATIVE NEWS - APPROACHES .pptx
COMMUNICATING NEGATIVE NEWS - APPROACHES .pptx
 
Sociology 101 Demonstration of Learning Exhibit
Sociology 101 Demonstration of Learning ExhibitSociology 101 Demonstration of Learning Exhibit
Sociology 101 Demonstration of Learning Exhibit
 
Mehran University Newsletter Vol-X, Issue-I, 2024
Mehran University Newsletter Vol-X, Issue-I, 2024Mehran University Newsletter Vol-X, Issue-I, 2024
Mehran University Newsletter Vol-X, Issue-I, 2024
 
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
 
Key note speaker Neum_Admir Softic_ENG.pdf
Key note speaker Neum_Admir Softic_ENG.pdfKey note speaker Neum_Admir Softic_ENG.pdf
Key note speaker Neum_Admir Softic_ENG.pdf
 
Micro-Scholarship, What it is, How can it help me.pdf
Micro-Scholarship, What it is, How can it help me.pdfMicro-Scholarship, What it is, How can it help me.pdf
Micro-Scholarship, What it is, How can it help me.pdf
 
NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...
NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...
NO1 Top Black Magic Specialist In Lahore Black magic In Pakistan Kala Ilam Ex...
 
Unit 3 Emotional Intelligence and Spiritual Intelligence.pdf
Unit 3 Emotional Intelligence and Spiritual Intelligence.pdfUnit 3 Emotional Intelligence and Spiritual Intelligence.pdf
Unit 3 Emotional Intelligence and Spiritual Intelligence.pdf
 

Unit2.polynomials.algebraicfractions

  • 1. F. Cano Cuenca 1 Mathematics 4º ESO Unit 2: POLYNOMIALS. ALGEBRAIC FRACTIONS 2.1.- DIVISION OF POLYNOMIALS Division of monomials To divide a monomial by a monomial, divide the numerical coefficients and then subtract the exponents of the same variables. Examples: 5 2 3 15x 3x 5x = − − 3 3 6x 6 55x = 4 6 3 2 4 12x y 4x y 3xy = Division of polynomials The division of polynomials is similar to the division of natural numbers. When you divide polynomials you get a quotient and a remainder. Example: 4 2 A(x) 6x 8x 7x 40= + + + 2 B(x) 2x 4x 5= − + The quotient is 2 17 Q(x) 3x 6x 2 = + + and the remainder is 5 R(x) 11x 2 = − . ( )4 2 2 2 17 5 6x 8x 7x 40 2x 4x 5 3x 6x 11x 2 2     + + + = − + ⋅ + + + −        4 2 6x 8x 7x 40+ + + 2 2x 4x 5− + 4 3 2 6x 12x 15x− + − 2 17 3x 6x 2 + + 3 2 12x 24x 30x− + − 2 85 17x 34x 2 − + − 5 11x 2 − 3 2 12x 7x 7x− + 2 17x 23x 40− +
  • 2. F. Cano Cuenca 2 Mathematics 4º ESO In general, if you divide the polynomial A(x) by the polynomial B(x) and the quotient and the remainder are Q(x) and R(x) respectively, you can write that: A(x) B(x) Q(x) R(x)= ⋅ + . When the remainder is 0, we have that A(x) B(x) Q(x)= ⋅ . In this case, the polynomial A(x) is divisible by B(x), that is, B(x) is a factor or divisor of A(x). Exercise 1: Work out the following divisions of polynomials. a) ( ) ( )5 4 3 2 x 7x x 8 : x 3x 1− + − − + b) ( ) ( )5 4 3 2 2 4x 20x 18x 28x 28x 6 : x 5x 3+ − − + − + − c) ( ) ( )4 2 2 6x 3x 2x : 3x 2+ − + d) ( ) ( )2 3 2 45x 120x 80x : 3x 4+ + + Division of a polynomial by x-a. Ruffini’s rule It is very common to divide a polynomial by x a− . For example: The quotient is 3 2 7x 10x 30x 4+ + − and the remainder is 5− . ( )( ) ( )4 3 3 2 7x 11x 94x 7 x 3 7x 10x 30x 4 5− − + = − + + − + − A(x) B(x) Q(x)R(x) − − +4 3 7x 11x 94x 7 x 3− − +4 3 7x 21x + + −3 2 7x 10x 30x 4 − +3 2 10x 30x − +2 30x 90x − +4x 7 3 10x −2 30x 94x −4x 12 −5
  • 3. F. Cano Cuenca 3 Mathematics 4º ESO But this division can also be done using Ruffini’s rule: We start writing the coefficients of the dividend and the number a. QUOTIENT: 7 10 30 4− , that is, 3 2 7x 10x 30x 4+ + − REMAINDER: 5− Notice that Ruffini’s rule’s steps are exactly the same as the steps of the long division. The advantage of Ruffini’s rule is that you only work with the coefficients and only do the essential operations. IMPORTANT!! Exercise 2: Use Ruffini’s rule for doing the following divisions of polynomials. a) ( ) ( )4 2 5x 6x 11x 13 : x 2+ − + − b) ( ) ( )5 4 6x 3x 2x : x 1− + + c) ( ) ( )4 3 2 3x 5x 7x 2x 13 : x 4− + − + − d) ( ) ( )4 3 2 6x 4x 51x 3x 9 : x 3+ − − − + QUOTIENT ’S COEFFICIENTS REMAINDER Ruffini’s rule only works when you divide a polynomial by a linear factor x a− .
  • 4. F. Cano Cuenca 4 Mathematics 4º ESO 2.2.- RUFFINI’S RULE’S USES Look at the division ( ) ( )3 2 2x 8x 31x 42 : x 6− − + − 2 -8 -31 42 6 12 24 ( 7) 6− ⋅ 2 4 -7 0 The quotient is 2 2x 4x 7+ − and the remainder is 0. Therefore, you can write that ( )( )3 2 2 2x 8x 31x 42 x 6 2x 4x 7− − + = − + − . Then, ( )x 6− is a factor of the polynomial 3 2 2x 8x 31x 42− − + , that is, the polynomial 3 2 2x 8x 31x 42− − + is divisible by ( )x 6− . Notice that 6 is a divisor of 42. So if you are looking for factors of a polynomial P(x), have a try with the linear factors (x a)− where “a” is a divisor of the constant term of P(x). Exercise 3: Find two linear factors of the polynomial 4 3 2 x 3x 2x 10x 12+ − − − . Exercise 4: Check if the following polynomials are divisible by x 3− or x 1+ . a) 3 2 A(x) x 3x x 3= − + − b) 3 2 B(x) x 4x 11x 30= + − − c) 4 3 2 C(x) x 7x 5x 13= − + − The Remainder Theorem Remember that you can calculate the number value of a polynomial at a given value of the variable. When the coefficients of a polynomial P(x) are integers, if(x a)− is a factor of P(x) and “a” is also an integer number, then “a” is a divisor of the constant term of P(x).
  • 5. F. Cano Cuenca 5 Mathematics 4º ESO Example: Calculate the number value of 3 2 P(x) 2x x 4x 2= − − + at x 3= − ( ) ( ) ( ) ( ) 3 2 P( ) 2 4 2 2 27 9 123 3 3 2 54 9 12 2 493= − − + = ⋅ − − + + = − − + +− − = −− − The Remainder Theorem states: Proof: If x a P(a) (a a) Q(a) R R= ⇒ = − ⋅ + = P(a) R⇒ = Exercise 5: Use Ruffini’s rule to calculate P(a) in the following cases. a) 4 2 P(x) 7x 5x 2x 24= − + − , a 5= − , a 10= b) 3 2 P(x) 3x 8x 3x= − + , a 1= , a 8= Exercise 6: Find the value of m so that the polynomial 3 2 P(x) x mx 5x 2= − + − is divisible by x 1+ . Exercise 7: The remainder of the division ( ) ( )4 3 2x kx 7x 6 : x 2+ − + − is 8− . What is the value of k? 2.3.- FACTORIZING POLYNOMIALS Roots of a polynomial A number “a” is called a root of a polynomial P(x) if P(a) 0= . The roots (or zeroes) of a polynomial are the solutions of the equation P(x) 0= . Examples: a) The numbers 1 and 1− are roots of the polynomial 2 P(x) x 1= − . 2 P(1) 1 1 0= − = 2 P( 1) ( 1) 1 0− = − − = b) Find the roots of the polynomial 2 P(x) x 5x 6= − + . The number value of the polynomial P(x) at x a= is the same as the remainder of the division P(x) : (x a)− . That is, P(a) R= . P(x) (x a) Q(x) R= − ⋅ +
  • 6. F. Cano Cuenca 6 Mathematics 4º ESO The roots of P(x) are the solutions of the equation P(x) 0= . 2 1 2 x 2 P(x) 0 x 5x 6 0 x 3  = = ⇒ − + = ⇒  = The roots of the polynomial 2 P(x) x 5x 6= − + are 2 and 3. One of the most important uses of Ruffini’s rule is to find the roots of a polynomial. Remember that the remainder of the division P(x) : (x a)− is the same as P(a). Therefore, if the remainder is 0, then P(a) 0= , so the number “a” is a root of P(x). Examples: Find a root of the polynomial 4 3 2 P(x) x 2x 7x 8x 12= + − − + 1 2 -7 -8 12 1 1 3 -4 -12 1 3 -4 -12 0 The remainder of the division P(x) : (x 1)− is 0. Then, 1 is a root of P(x). An interesting theorem about polynomials is the Fundamental Theorem of Algebra: Let n n 1 n 2 2 n n 1 n 2 2 1 0 P(x) a x a x a x ... a x a x a− − − − = + + + + + + be any polynomial of degree n. If 1 2 3 n r, r , r ,...,r are its n roots, then the polynomial can be factorized as: Example: Find the roots of the polynomial 2 P(x) 2x x 1= − − and factorize it. 1 2 2 x 1 P(x) 0 2x x 1 0 1 x 2  =  = ⇒ − − = ⇒  = −  If the remainder of the division P(x) : (x a)− is 0, then the number “a” is a root of P(x). Any polynomial of degree n has n roots. (These roots can be real or complex numbers). n 1 2 n 1 n P(x) a (x r) (x r ) .... (x r ) (x r )− = − ⋅ − ⋅ ⋅ − ⋅ −
  • 7. F. Cano Cuenca 7 Mathematics 4º ESO The roots of 2 P(x) 2x x 1= − − are 1 and 1 2 − . The factorization of P(x) is ( ) ( )1 1 P(x) 2 x 1 x 2 x 1 x 2 2      = − − − = − +          . Factorization techniques You have seen before the factorization of a polynomial in the previous example. Factorize a polynomial means to write it as a product of lower degree polynomials. Examples: ( )( )2 x 9 x 3 x 3− = + − ( )( )3 2 x 5x 6x x x 2 x 3− + = − − There are different techniques for factorizing polynomials: 1) Taking out common factor. Example: ( )2 8x 2x 2x 4x 1− = − 2) Using polynomial identities (the square of the sum, the square of the difference, the product of a sum and a difference). Examples: ( ) ( )( ) 22 x 10x 25 x 5 x 5 x 5+ + = + = + + ( ) ( )( ) 22 x 6x 9 x 3 x 3 x 3− + = − = − − ( )( )2 x 16 x 4 x 4− = + − 3) Using the Fundamental Theorem of Algebra. Example: The roots of the polynomial 2 P(x) x x 6= − − + are 2 and 3− . ( )( )( )2 P(x) x x 6 1 x 2 x 3= − − + = − − + Exercise 8: Take out common factor or use the polynomial identities to factorize the following polynomials. a) 2 3x 12x− b) 3 2 4x 24x 36x− + c) 2 4 45x 5x− d) 4 2 3 x x 2x+ + e) 6 2 x 16x− f) 4 16x 9− Exercise 9: Use the Fundamental Theorem of Algebra to factorize the following polynomials. a) 2 x 4x 5+ − b) 2 x 8x 15+ + c) 2 7x 21x 280− − d) 2 3x 9x 210+ −
  • 8. F. Cano Cuenca 8 Mathematics 4º ESO 4) Using Ruffini’s rule. Example: 3 2 P(x) x 2x 5x 6= − − + ( )( ) ( )( )( )3 2 2 P(x) x 2x 5x 6 x 1 x x 6 x 1 x 2 x 3= − − + = − − − = − + − 5) A combination of the previous ones. Examples: a) ( ) ( ) 25 4 3 3 2 3 P(x) 12x 36x 27x 3x 4x 12x 9 3x 2x 3= − + = − + = − If we solve the equation P(x) 0= , we get the roots of P(x). ( ) ( ) ( ) 23 P(x) 0 3x 2x 3 0 3 x x x 2x 3 2x 3 0= ⇒ − = ⇒ ⋅ ⋅ ⋅ ⋅ − ⋅ − = ⇒ 1 2 3 4 5 3 3 x 0, x 0, x 0, x , x 2 2 ⇒ = = = = = The polynomial has a double root and a triple root. b) 3 P(x) x x 6= − − Firstly, we use Ruffini’s rule: Secondly, we find the roots of the polynomial 2 x 2x 3− + . 1 -2 -5 6 1 1 -1 -6 1 -1 -6 0 -2 -2 6 1 -3 0 Taking out common factor Using polynomial identities 1 0 -1 6 -2 -2 4 -6 1 -2 3 0
  • 9. F. Cano Cuenca 9 Mathematics 4º ESO 2 x 2x 3 0− + = a 1, b 2, c 3= = − = ( ) 2 2 2 4 1 3 2 8 x 2 1 2 ± − − ⋅ ⋅ ± − = = ⋅ The roots of the polynomial 2 x 2x 3− + are not real numbers. So the factorization of P(x) is ( )( )2 P(x) x 2 x 2x 3= + − + . Exercise 10: Factorize the following polynomials. a) 2 3x 2x 8+ − b) 5 3x 48x− c) 3 2 2x x 5x 12+ − + d) 3 2 x 7x 8x 16− + + e) 4 3 2 x 2x 23x 60x+ − − f) 4 3 2 9x 36x 26x 4x 3− + + − Exercise 11: Take out common factor in the following expressions. a) 3x(x 3) (x 1)(x 3)− − + − b) (x 5)(2x 1) (x 5)(2x 1)+ − + − − c) (3 y)(a b) (a b)(3 y)− + − − − Exercise 12: Write second degree polynomials whose roots are: a) 7 and -7 b) 0 and 5 c) -2 and -3 d) 4 (double) Exercise 13: Prove that the polynomial n x 1− is divisible by x 1− for any value of n. Find the general expression for the quotient of this division. Exercise 14: The remainder of the division 2 P(x) : (x 1)− is 4x 4+ . Find the remainder of the division P(x) : (x 1)− . Exercise 15: Prove that the polynomial 2 x (a b)x ab+ + + is divisible by x a+ and by x b+ for any values of a and b. Find its factorization. 2 8i 2 2 2i2 8 1 2i 2 2 2 + ++ − = = = + ∉ » 2 8i 2 2 2i2 8 1 2i 2 2 2 − −− − = = = − ∉ »
  • 10. F. Cano Cuenca 10 Mathematics 4º ESO 2.4.- ALGEBRAIC FRACTIONS An algebraic fraction is the quotient of two polynomials, that is, P(x) Q(x) Examples: 2 2x 4 x 5x 3 + − + 1 x 5− 4 3 2 x 5x 1 x x − + + 2x 5 11 + The same calculations that you do with numerical fractions can be done with algebraic fractions. Simplification of algebraic fractions To simplify algebraic fractions: • Factorize the polynomials. • Cancel out the common factors. Example: ( )2 2 x 1x 1 x 2x 3 +− = − − ( ) ( ) x 1 x 1 − + ( ) x 1 x 3x 3 − = −− Addition, subtraction, product and division of algebraic fractions You can add, subtract, multiply an divide algebraic fractions in the same way that you do in simple arithmetic. Examples: a) ( ) ( )( ) ( ) ( ) ( ) ( ) 2 22 x 7 x 2 2x 1 x x 1 x 1 x 7 x 2 x 2x 1 2x 7x 9 x x 1x 1 x x 1x + + −+ − − − + − + + + + = + = + + + b) ( )( ) ( ) 2 2 3 2 3 222 x 2 x 1 xx x 2x 2 x x 2 2x x 2 2 x 1 2x 1 xx 2 1x xx + − − + −+ − + − = = = + ++ ⋅ + c) ( )( ) ( )( ) ( ) ( ) 2 2 x 5 x x 2x 5 x 2x x x 5 2x 1 : x 2 x 2x 12x 2 + ++ + = = ++ + + − + ( )x 2+ ( ) ( ) 2x 5 x x 5x 2x 1 2x 12x 1 + + = = + ++ d) ( )( ) ( ) ( )( )( ) ( ) ( ) 2 2 2 2 2 2 x 9 xx 9 x 4 x 9 x 4 x 2x 3 : x 2x 2 4 x 3 : x x x2 x x 2 x2 x 3 − − − − −− = = = −+− + − − ⋅ − − + ( ) ( )x 3 x 3+ − = ( )x 2+ ( )( ) ( ) x 2 x 2 x 2 − − + ( )x x 3− 3 2 x x 8x 12 x − − + = ( )LCM x, x(x 1), (x 1) x(x 1)− − = −
  • 11. F. Cano Cuenca 11 Mathematics 4º ESO Exercise 16: Simplify the following algebraic fractions. a) 2 3 2x 6x 4x 2x − − b) ( ) ( ) ( ) ( ) 2 2 x 3 x x 3 x 3 x x 2 − + − + c) 3 2 3 2 x 3x x 3 x 3x + + + + d) 3 2 3 2 x 5x 6x x x 14x 24 − + − − + Exercise 17: Work out and simplify. a) 2 2 2x 1 x 5 x 3 x 3x + + − + + b) 2 2 3 x x x x 1 x 1   −  + −  c) 2 5x 10 x 9 x 3 x 2 − − ⋅ + − d) 2 3x 1 x 3 2x 5 x x 2x 2x − + + − + −− e) 2 2x 1 x : 2x 1 4x 2 + − − f) 2 x 1 1 : x 1 x x 1   −  − −  Exercise 18: Translate into algebraic language. (You can only use one unknown). a) The quotient between two even consecutive numbers. b) A number minus its inverse. c) The inverse of a number plus the inverse of twice that number. d) The sum of the inverses of two consecutive numbers. Exercise 19: Use polynomials to express the area and the volume of this prism. Exercise 20: A tap takes x minutes to fill a tank. Another tap takes 3 minutes less than the first one to fill the same tank. Express as a function of x the part of the tank to be filled if we turn both taps on for one minute.
  • 12. F. Cano Cuenca 12 Mathematics 4º ESO Exercise 21: We mix “x” kg of a paint that costs 5 €/kg with “y” kg of another paint that costs 3 €/k. Express the price of 1 kg of the mixture as a function of “x” and “y”. Exercise 22: Two cyclists start at the same time from opposite ends of a course that is 60 miles long. One cyclist is riding at x mph and the second cyclist is riding at x 3+ mph. How long after they begin will they meet? (Give the answer as a function of x). Exercise 23: A rhombus is inscribed in a rectangle whose sides are x an y. Express the perimeter of the rhombus as a function of x and y. Exercise 24: The sides AB and BC in the rectangle ABCD are 3 and 5 cm respectively. If AA' BB' CC' DD' x= = = = , express the area of the rhomboid A’B’C’D’ as a function of x. Exercise 25: The number value of the polynomial 34 x 3 π express the volume of a sphere with radius x, and the number value of the polynomial 2 4 xπ express the area of a sphere with radius x. a) Is there any sphere whose volume (expressed in m3 ) is the same as its area (expressed in m2 )? If your answer is affirmative, find the radius of that sphere. b) Find the relation between the area of a sphere and the area of a maximum circle in that sphere. c) Use a polynomial of x to express the volume of the cylinder circumscribed in a sphere with radius x. Find the relation between this polynomial and the one which express the volume of the sphere.