es de suma importancia

1. RAICES DE UN POLINOMIO,
FACTORIZACION
1.- CALCULAR LAS RAÍCES DE ESTOS
POLINOMIOS
A) 𝑷( 𝑿) = 𝑿 𝟑 − 𝟑𝑿 𝟐 + 𝟐
𝑋3 − 3𝑋2 + 0𝑋 + 2
1
0 -1 1
0 1 1 X=1
1 1 0 0
𝑥 =
−(−2) ± √(−2)2 − 4(1)(−2)
2(1)
𝑋 =
2 ± √12
2
𝑋 =
2 ± √4 ∗ 3
2
𝑋 =
2 ± 2√3
2
𝑋1 =
2 + 2√3
2
=
2
2
+
2√3
2
= 1 + √3
𝑋2 =
2 − 2√3
2
=
2
2
−
2√3
2
= 1 − √3
𝐶. 𝑆 = (1 + √3) (1 − √3)
B) 𝑿 𝟐 − 𝟐𝑿 + 𝟏
X2
-2X 1
X -1 = -1X
X -1 = -1X
-2X
𝐶. 𝑆 = ( 𝑋 = 1) ( 𝑋 = 1)
C) 𝑹( 𝑿) = 𝑿 𝟑 − 𝟐𝑿 𝟐 − 𝟓𝑿 − 𝟔
1 -2 -5 6
1 -1 -6 X=+1
1 -1 -6 0
𝑥 =
−(−2) ± √(−2)2 − 4(1)(−6)
2(1)
𝑋 =
2 ± √4 + 24
2
=
2 ± √28
2
𝑥 =
2 ± √4 ∗ 7
2
=
2 ± 2√7
2
𝑋1 =
2 + 2√7
2
=
2
2
+
2√7
2
= 1 + √7
𝑋2 =
2 − 2√7
2
=
2
2
−
2√7
2
= 𝟏 − √𝟕
D) 𝑺( 𝑿) = 𝑿 𝟐 − 𝟓𝑿 − 𝟏𝟒
𝑥 =
−(−5) ± √(−5)2 − 4(1)(−14)
2(1)
𝑥 =
5 ± √25 + 56
2
=
5 ± √81
2
=
5 ± 9
2
𝑋1 =
5 + 9
2
=
14
2
= 7
𝑋2 =
5 − 9
2
=
−4
2
= −2
𝐶. 𝑆 = ( 𝑋 = 7)(𝑋 = −2)
2.- CALCULAR LAS RAÍCES DE ESTOS
POLINOMIOS.
A) 𝑿 𝟑 − 𝟗𝑿 𝟐 + 𝟐𝟔𝑿 − 𝟐𝟒
1
0 -1 1
0 1 1 X=1
1 1 0 0
X=1
1 3 0
4 X=4
1 0
𝐶. 𝑆 = ( 𝑋 = 3), ( 𝑋 = 2),(𝑋 =

2. 𝟒) B) 𝑿 𝟑 − 𝟐𝑿 𝟐 − 𝟑𝑿
𝑋3 − 2𝑋2 − 3𝑋 + 0
1
0 -1 1
0 1 1 X=1
1 1 0 0
𝑥 =
−(−2) ± √(−2)2 − 4(1)(−3)
2(1)
𝑥 =
2 ± √4 + 12
2
=
2 ± √18
2
𝑥 =
2 ± √9 ∗ 2
2
=
2 ± 3√2
2
𝑋1 =
2 + 3√2
2
=
2
2
+
3√2
2
= 1 +
3√2
2
𝑋2 =
2 − 3√2
2
=
2
2
−
3√2
2
= 1 −
3√2
2
𝑪. 𝑆 = ( 𝑋 = 0),(1 +
3√2
2
), (1 −
3√2
2
)
C.- 𝑿 𝟒 − 𝑿 𝟐 − 𝑿 + 𝟏
𝑋4 − 0𝑋3 − 𝑋2 − 𝑋 + 1
1 0 -1 1
0 1 1 X=1
1 1 0 0
𝑥 =
−1 ± √12 − 4(1)(0)
2(1)
=
−1 ± √1
2
𝑋1 =
−1 + √1
2
𝑋2 =
−1 − √1
2
𝐶. 𝑆:( 𝑋 = 1), ( 𝑋 =
−1 + √1
2
) ( 𝑋 =
−1 − √1
2
)
𝑫) 𝑿 𝟑 + 𝑿 𝟐 − 𝟗𝑿 − 𝟗
1 1 -9 -9
-1 0 9
X=-
1
1 0 -9 0
X2
OX -9
X 3 = -3X
X -3 = 3X
0X
𝐶. 𝑆:( 𝑋 = −1), ( 𝑋 = −3),( 𝑋 = 3)
𝑬) 𝑿 𝟐 − 𝑿 − 𝟐
X2
-X -2
X 1 = -2X
X -2 = 1X
-1X
𝐶. 𝑆:( 𝑋 = −1), ( 𝑋 = 2)
𝑭) 𝑿 𝟐 + 𝑿
X2
X 0
X 0 = 1X
X 1 = 0X
1X
𝐶. 𝑆:( 𝑋 = −1), (𝑋 = 0)
𝑮) 𝟒𝑿 𝟐 − 𝟏𝟔
4X2
0X -16
2X -4 = 8X
2X 4 = -8X
0X
𝐶. 𝑆:( 𝑋 = −2), ( 𝑋 = 2)
𝑭) 𝑿 𝟐 − 𝟒𝑿 + 𝟒
X2
-4X 4
X -2 = -2X
X -2 = -2X
-4X
𝐶. 𝑆:( 𝑋 = 2), ( 𝑋 = 2)
3.- FACTORIZAR ESTOS POLINOMIOS,
APLICANDO LAS IGUALDADES
NOTABLES.
𝑨) 𝑿 𝟐 + 𝟐𝑿 + 𝟏
X2
2X 1

3. X 1 = 1X
X 1 = 1X
2X
𝐶. 𝑆:( 𝑋 = −1), ( 𝑋 = −1)
𝑩) 𝑿 𝟐 + 𝟏𝟎𝑿 + 𝟐𝟓
X2
10X 25
X 5 = 5X
X 5 = 5X
10X
𝐶. 𝑆:( 𝑋 = −5), ( 𝑋 = −5)
𝑪) 𝑿 𝟐 − 𝟒
𝑋2 + 0𝑋 − 4
X2
0X -4
X -2 = 2X
X 2 = -2X
0X
𝐶. 𝑆:( 𝑋 = 2), ( 𝑋 = −2)
𝑫) 𝟒𝑿 𝟐 − 𝟏𝟔
4𝑋 𝟐 + 0𝑋 − 16
4X2
0X -16
X -4 = 4X
X 4 = -4X
0X
𝐶. 𝑆:( 𝑋 = −4), ( 𝑋 = 4)
𝑬) 𝑿 𝟑 − 𝟗𝑿 𝟐+27X-27
1 -9 27 -27
3 -18 27 X=3
1 -6 9 0
X2
-6X 9
X -3 = -3X
X -3 = -3X
-6X
𝐶. 𝑆:( 𝑋 = 3), ( 𝑋 = 3), ( 𝑋 = 3)
4.- FACTORIZA ESTOS POLINOMIOS
𝑨) 𝑿 𝟐 + 𝟓𝑿 + 𝟔
X2
5X 6
X 2 = 3X
X 3 = 2X
5X
𝐶. 𝑆:( 𝑋 = −2), ( 𝑋 = −3)
𝑩) 𝑿 𝟐 + 𝑿 − 𝟏𝟐
X2
X -12
X -3 = 4X
X 4 = -3X
1X
𝐶. 𝑆:( 𝑋 = 3), ( 𝑋 = −4)
𝑪) 𝑿 𝟐 + 𝟏𝟏𝑿 + 𝟐𝟒
X2 11X 24
X 3 = 8X
X 8 = 3X
11X
𝐶. 𝑆: (X= -3), (X= -8)
𝑫) 𝑿 𝟑 − 𝟏𝟑𝑿 + 𝟏𝟐
𝑋3 + 0𝑋2 − 13𝑋 + 12
1 0 -13 12
1 1 -12 X=1
1 1 -12 0
X2
1X -12
X -3 = 4X
X 4 = -3X
1X
𝐶. 𝑆:( 𝑋 = 1), ( 𝑋 = −4), ( 𝑋 = 3)

4. 𝑬) 𝑿 𝟑 − 𝟓𝑿 𝟐 − 𝑿 + 𝟓
1 -5 -1 5
-1 6 -5
X=-
1
1 -6 5 0
X2
-6X 5
X -1 = -5X
X -5 = -1X
-6X
𝐶. 𝑆:( 𝑋 = −1), ( 𝑋 = 5), ( 𝑋 = 1)
𝑭) 𝑿 𝟑 + 𝟒𝑿 𝟐-11X-30
1 4 -11 -30
-2 -4 30
X=-
2
1 2 -15 0
X2
2X -15
X -3 = 5X
X 5 = -3X
2X
𝐶. 𝑆:( 𝑋 = −2), ( 𝑋 = 3), ( 𝑋 = −5)
5.- FACTORIZA ESTOS POLINOMIOS
𝟐𝑿 𝟑 − 𝟖𝑿 𝟐 + 𝟐𝑿 + 𝟏𝟐
2 -8 2 12
4 -8 -12 X=2
2 -4 -6 0
2X2
-4X -6
2X -6 = 2X
X 1 = -6X
-4X
𝐶. 𝑆:( 𝑋 = 2), ( 𝑋 = −
1
2
), ( 𝑋 = 6)
𝑩) 𝟑𝑿 𝟑 − 𝟖𝑿 𝟐 − 𝟐𝟎𝑿 + 𝟏𝟔
3 -8 -20 16
12 16 -16 X=2
3 4 -4 0
3X2
4X -4
3X -2 = 6X
X 2 = -2X
4X
𝐶. 𝑆;( 𝑋 = 2), ( 𝑋 = −
2
3
), ( 𝑋 = 2)
𝑪) 𝟐𝑿 𝟒 + 𝟏𝟓𝑿 𝟑 + 𝟑𝟏𝑿 𝟐 + 𝟏𝟐𝑿
2𝑋4 + 15𝑋3 + 31𝑋2 + 12𝑋 + 0
2 15 31 12 0
0 0 0 0 X=0
2 15 31 12 0
-6 -27 -12
X=-
3
2 9 4 0
2X2
9X 4
2X 1 = 8X
X 4 = 1X
9X
6.- DESCOMPON EN FACTORES ESTOS
POLINOMIOS Y DI CUALES SON SUS
RAICES:
𝑨) 𝑿 𝟑 − 𝟐𝑿 𝟐 − 𝑿 + 𝟐
1 -2 -1 2
1 -1 -2 X=1
1 -1 -2 0
X2
-1X -2
X 1 = -2X
X -2 = 1X
-1X
𝐶, 𝑆:( 𝑋 = 1), ( 𝑋 = 2), ( 𝑋 = −1)

5. 𝑩) 𝑿 𝟒 − 𝟓𝑿 𝟐 + 𝟒
𝑋4 + 0𝑋3 − 5𝑋2 + 0𝑋 + 4
1 0 -5 0 4
1 1 -4 -4 X=1
1 1 -4 -4 0
-2 2 4
X=-
2
1 -1 -2 0
X2
-1X -2
X 1 = -2X
X -2 = 1X
-1X
𝐶. 𝑆:( 𝑋 = 1), ( 𝑋 = −2), ( 𝑋 = 2),( 𝑋 = −1)
𝑪) 𝟐𝑿 𝟑 − 𝟑𝑿 𝟐 − 𝟗𝑿 + 𝟏𝟎
2 -3 -9 10
2 -1 -10 X=1
2 -1 -10 0
2X2
-1X -10
2X -5 = 4X
X 2 = -5X
-1X
𝐶. 𝑆:( 𝑋 = 1), ( 𝑋 = −1), ( 𝑋 = 5)
𝑫) 𝑿 𝟓 − 𝟕𝑿 𝟒 + 𝟏𝟎𝑿 𝟑 − 𝑿 𝟐 + 𝟕𝑿 − 𝟏𝟎
1 -7 10 -1 7 -10
2 -10 0 -2 10 X=2
1 -5 0 -1 5 0
1 -4 -4 -5
X=
1
1 -4 -4 -5 0
5 5 5 X=5
1 1 1 0
𝑥 =
−1 ± √12 − 4(1)(1)
2(1)
=
−1 ± √−3
2
𝑋1 =
−1 + √−3
2
𝑋2 =
−1 − √−3
2
𝐶. 𝑆:( 𝑋 = 2), ( 𝑋 = 1), ( 𝑋 = 5),
( 𝑋 =
−1 + √−3
2
) ,(
−1 − √−3
2
)
𝑬) 𝟔𝑿 𝟒 − 𝟓𝑿 𝟑 − 𝟐𝟑𝑿 𝟐 + 𝟐𝟎𝑿 − 𝟒
6 -5 -23 20 -4
12 14 -18 4 X=2
6 7 -9 2 0
-12 8 -2
X=-
2
6 -4 -1 0
𝑥 =
−(−4) ± √(−4)2 − 4(6)(−1)
2(−1)
𝑋 =
4 ± √16 + 24
−2
=
4 ± √40
−2
=
4 ± 2√10
−2
𝑋1 =
4 + 2√10
−2
= −
4
2
−
2√10
2
= −2 − √10
𝑋2 =
4 − 2√10
−2
= −
4
2
+
2√10
2
= −2 + √10
𝐶. 𝑆:( 𝑋 = 2), ( 𝑋 = −2)
( 𝑋 = −2 − √10),( 𝑋 = −2 + √10)
𝑭) 𝑿 𝟓 − 𝟏𝟔𝑿
𝑋5 + 0𝑋4 + 0𝑋3 + 0𝑋2 + 16𝑋 + 0
1 0 0 0 -16 0
0 0 0 0 0 X=0
1 0 0 0 -16 0
2 4 8 16
X=
2
1 2 4 8 0
-2 0 -8
X=-
2
1 6 4 0
𝑥 =
−6 ± √62 − 4(−1)(4)
2(1)
=
−6 ± √36 + 16
2

6. 𝑋 =
−6 ± √4 ∗ 13
2
=
−6 ± 2√13
2
𝑋1 =
−6 + 2√13
2
=
−6
2
+
2√13
2
= −3 + √13
𝑋2 =
−6 − 2√13
2
=
−6
2
−
2√13
2
= −3 − √13
𝐶. 𝑆:( 𝑋 = 0), ( 𝑋 = 2), ( 𝑋 = −2),
( 𝑋 = −3 + √13),(𝑋 = −3 − √13)
𝑮) 𝟒𝑿 𝟐 − 𝟐𝟓
4𝑋2 + 0𝑋 − 25
4X2
0X 25
2X -5 = -10X
2X 5 = 10X
0X
𝐶. 𝑆:( 𝑋 = −
5
2
) ,( 𝑋 =
5
2
)
7.- FACTORIZA LOS SGTES POLINOMIOS
Y CALCULA SUS RAICES.
𝑨) 𝑿 𝟑 − 𝟐𝑿 𝟐 − 𝟓𝑿 + 𝟔
1 -2 -5 6
1 -1 -6 X=1
1 -1 -6 0
X2
-1X -6
X 2 = -3X
X -3 = 2X
-1X
𝐶. 𝑆:( 𝑋 = 1), ( 𝑋 = 3), ( 𝑋 = −2)
𝑩) 𝑿 𝟑 − 𝟓𝑿 𝟐 + 𝟕𝑿 − 𝟑
1 -5 7 -3
1 -4 3 X=1
1 -4 3 0
X2
-4X 3
X -1 = -3X
X -3 = -1X
-4X
𝐶. 𝑆:( 𝑋 = 1), ( 𝑋 = 3), ( 𝑋 = 1)
𝑪) 𝑿 𝟒 − 𝟗𝑿 𝟐 + 𝟒𝑿 + 𝟏𝟐
𝑋4 + 0𝑋3 − 9𝑋2 + 4𝑋 + 12
1 0 -9 4 12
2 4 -10 -12 X=2
1 2 -5 -6 0
-3 3 6
X=-
3
1 -1 -2 0
X2
-1X -2
X 1 = -2X
X -2 = 1X
-1X
𝐶. 𝑆:( 𝑋 = 2), ( 𝑋 = −3), ( 𝑋 = 2),(𝑋 = −1)
𝑫) 𝑿 𝟒 − 𝟖𝑿 𝟑 + 𝟏𝟒𝑿 𝟐 + 𝟖𝑿 − 𝟏𝟓
1 -8 14 8 -15
3 -15 -3 15 X=3
1 -5 -1 5 0
1 -4 -5 X= 1
1 -4 -5 0
X2
-4X -5
X 1 = -5X
X -5 = 1X
-4X
𝐶. 𝑆:( 𝑋 = 3), ( 𝑋 = 1), ( 𝑋 = 5),( 𝑋 = −1)