This document provides examples of factorizing polynomials by: 1. Finding the roots of polynomials using the quadratic formula. 2. Factoring polynomials using the difference of squares and perfect square trinomial identities. 3. Factoring polynomials into irreducible factors. The document contains several examples of factorizing polynomials of varying degrees up to degree 5. The examples illustrate the process of finding the roots and then factorizing the polynomials based on those roots.

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)