The document provides examples of different types of mathematics exercises involving factoring polynomials and binomial expressions. These include factoring the difference and sum of squares, factoring perfect square trinomials, factoring the difference and sum of cubes, and factoring expressions involving squares, cubes, and binomials. Solutions are provided for factoring expressions in the forms of (x + a)(x + b), (x - a)(x + b), and (x - a)(x - b).

1. EJERCICIOS DE MATEMATICAS
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
=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+1) (a-1)
C) a2b3 – n4 + a2b3x2 – n4x2 – 3a2b3x + 3n4x
(a2b3-n4) = 1
(a2b3-n4)
(a2b3-n4x2) = 1-1x2
(a2b3-n4)
(-3a2b3x+3n4x) = -3x+3x
(a2b3-n4)
= (a2b3 - n4) (1-3x+x2)

2. 4) FACTORAR UN TRINOMIO CUADRADO PERFECTO
A) 1 + 25x4 – x2
25 36 3
Ordenando:
25x4 – x2 + 1
36 3 25
= 2
5x2 - 1
6 5
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)