This document provides examples of factorizing polynomial expressions in various forms: 1) Factoring the difference of two binomials. 2) Factoring the square of a binomial expression. 3) Factoring a trinomial into two binomial factors. 4) Factoring a perfect square trinomial. 5) Factoring the difference of squares. 6) Factoring a trinomial that is a perfect square by addition and subtraction. 7) Factoring quadratic trinomials in the form x^2 + bx + c. 8) Factoring a perfect cube of binomials. 9) Factoring the difference and sum of perfect cubes.

1. 1. PRODUCTO DE DOS BINOMIOS DE LA FORMA
(x+a) (x+b)
a) (ax+1 - 6) (ax+1 - 5)
=a2x+2 + 30 - 6a x+1 - 5a x+1
=a2x+2 - 11a x+1 + 30
2. EL CUADRADO DE UN MULTINOMIO
b) (2x3 - x2 + 4x - 3)2
= (2x3)2 + (-x2)2 + (4x)2 + (-3)2 + 2(2x3) (-x2) + 2(2x3) (4x) + 2(2x3) (-3)
+ 2(-x2) (4x) + 2(-x2) (-3)
+ 2(4x) (-3)
= 4x6 + x4 +16x2 + 9 – 4x5 +16x4 -12x3 – 8x3 + 6x2 -24x
= 4x6 - 4x5 +17x4 - 20x3 + 22x2 – 24x +9
3. FACTORAR O DESCOMPONER EN DOS
FACTORES
c) 24a2xy2 - 36x2y4
=6xy2 (4a2 -6xy2)
d) (a+3) (a+1) – 4 (a+1)
(a + 3) (a + 1) = (a + 3) 4(a + 1) = -4
(a + 1) (a + 1)

2. = (a + 1) (a + 3 – 4)
= (a + 1) (a – 1)
e) a2b3 – n4 + a2b3x2 – n4x2 – 3a2b3x + 3n4x
= ( a2b3 + a2b3x2 – 3a2b3x) (– n4 – n4x2 + 3n4x)
= a2b3 (1 + x2 - 3x) – n4 (1 + x2 – 3x)
= (1 + x2 - 3x) (a2b3 – n4 )
4. FACTORAR UN TRINOMIO CUADRADO
PERFECTO
f) 1 + 25x4 - x2
25 36 3
= 25x4 - x2 + 1
36 3 25
= (5 x2 - 1)2
6 5
5. DIFERENCIA DE CUADRADOS
g) x2 - y2z4
100 81

3. =( x + yz2) (x - yz2)
10 9 10 9
6. TRINOMIO CUADRADO PERFECTO POR
ADICIÓN Y SUSTRACCIÓN
h) 81m8 + 2m4 + 1
= 81m8 + 2m4 + 1
16m4 - 16m4
81m8 +18m4 + 1 - 16m4
= (9m4 + 1)2 - 16m4
= (9m4 + 1 - 4m2) (9m4 + 1 + 4m2)
= (9m4 - 4m2 + 1) (9m4 + 4m2 + 1)
7. TRINOMIO DE LA FORMA x2 + bx + c y ax2
+ bx + c
i) m2 - 2m - 168
= (m – 14) (m + 12)
j) 20n2 - 9n – 20
= 20(20n2) – 20(9n) – 20(20)

4. = 400n2 – 20(9n) – 400
= 20n2 – 25) (20n + 16)
20
= 20n2 – 25) (20n + 16)
5 4
= (4n – 5) (5n + 4)
8. CUBO PERFECTO DE BINOMIOS
k) 125x3 + 1 +75x2 + 15x
= 125x3 + 75x2 + 15x + 1
3(5x)2 (1)= 75x2
3(5x) (1)2 = 15x
= (5x + 1 )3
9. SUMA O DIFERENCIA DE CUBOS PERFECTOS
l) 27a3 – b3
= (3a - b) (3a) 2+ (3a) (b) + (b)2
= (3a – b) (9a2 + 3ab +b2)