This document provides examples of factoring polynomials of different types: 1) Factoring the difference of two squares. 2) Finding the square of a binomial expression. 3) Factoring trinomials and polynomials into two binomial factors. 4) Recognizing and factoring perfect square trinomials and differences of squares. The document demonstrates various factoring techniques for polynomials in one and multiple variables.

1. 1) PRODUCTO DE DOS BINOMIOS DE LA FORMA (X + A)(X + B)
A) (ax+1 – 6) (a x+1 – 5)
= a2x+2 – 5ax+1 - 6ax+1 + 30
= a2x+2 – 11ax+1 + 30
2) EL CUADRADO DE UN MULTINOMIO
A) (x – 2y + z – 3r)2
= (x)2+(-2y)2+(z)2+(-3r)2 +2(x)(-2y)-2(x)(z)+2(x)(-3r)
+2(-2y)(z)+2(-2y)(-3r) +2(z)(-3r)
= x2 - 4y2 + z2 – 9r2 – 4xy – 2xz – 6xr – 4yz + 12yr – 6zr
3) FACTORAR O DESCOMPONER EN DOS FACTORES
A) 24 a2xy2 – 36x2y4
Factor común: 12xy2
=12xy2(2a2-3xy2)
B) (a + 3) (a + 1) -4(a +1)
= (a + 3) (a + 1) = a + 3
(a + 1)
= - 4 (a +1) = -4
a+1
= a + 3 (a + 1) – 4 (a + 1) = (a + 1) (a + 3 - 4)
= (a + 1)(a – 1)

2. C) a2b3 – n4 + a2b3x2 – n4x2 – 3a2b3x + 3n4x
= (a2b3 - n4) (1-3x+x2)
4) FACTORAR UN TRINOMIO CUADRADO PERFECTO
A) 1 + 25x4 – x2
25 36 3
Ordenando:
1 – x2 + 25x4
25 3 36
2
= 1 - 5x2
5 6
5) DIFERENCIA DE CUADRADO
A) x2 - y2z4
100 81
= x + y z2 x - y z2
10 9
10 9
6) TRINOMIO CUADRADO POR ADICION Y SUSTRACION
A) 121 x4 – 133x2y4 + 36y8
11x2 6y4
2(11x2)(6y4) = 132 x2y4
= 121 x4 – 133x2y4 + 36y8
+ x2y4 - x2y4
121 x4 - 132 x2y4 + 36y8 - x2y4
= (121 x4 - 132 x2y4 + 36y8) - x2y4
= (11 x2 – 6y4)2 - x2y4
= 11 x2 – 6y4 - xy2
= (11x2 + xy – 6y4) (11x2 - xy – 6y4)

3. 7) TRINOMIO x2 + bx + x y ax2 + bx + x
A) c2 + 24c + 135
= (c +15 ) (c + 9)
B) 30x2 + 13x – 10
= 30(30x2)+ 30(13x) – 30(10)
= 900x2 +30(13x) - 300
= (30x + 25 ) (30x - 12)
5x6
= ( 6x + 5) (5x – 2)
8) CUBO PERFECTO BINOMIAL
(1 + 3a2 – 3a – a3)
= 1 – 3a + 3a2– a3
1 a
3(1) + (a)2 = 3a
3(1)2 + a = 3a2
= (1 - a)3
9) SUMA O DIFERENCIA DE CUBOS PERFECTOS
27a3 – b3
= (3a – b) (9a2 + 3ab + b2)