Explicación breve de la obtención del producto de binomios

1. PRODUCTOS NOTABLES
Cuadrado de la suma de un binomio
(5x + 1)2
=
(5x + 1)(5x + 1) =
(5x)(5x) + (5x)(1) + (1)(5x) + (1)(1) =
25x2
+ 5x + 5x + 1 = 25x2 + 10x + 1
(a + b)2
= a2 + 2ab + b2
(a + b)(a + b) =
(a)(a) + (a)(b) + (b)(a) + (b)(b) =
a2
+ ab + ab + b2
= a2
+ 2ab + b2
Cuadrado de la diferencia de un binomio
(6x – 3)2
=
(6x – 3)(6x – 3) =
(6x)(6x) + (6x)(– 3) + (– 3)(6x) + (– 3)(– 3) =
36x2
– 18x – 18x + 9 = 36x2 – 36x + 9
(a – b)2
= a2 – 2ab + b2
(a – b)(a – b) =
(a)(a) + (a)(– b) + (– b)(a) + (– b)(b) =
a2
– ab – ab + b2
= a2
– 2ab + b2
Producto de un binomio conjugado
(2x + 3)(2x – 3) =
(2x)(2x) + (2x)(– 3) + (3)(2x) + (3)(– 3) =
4x2
– 6x + 6x – 9 = 4x2 – 9
(a + b)(a – b) = a2 – b2
(a)(a) + (a)(– b) + (b)(a) + (b)(– b) =
a2
– ab + ab – b2
=
a2
– b2

2. Producto de binomios con un término común
(2x + 5)(2x + 3) = CASO 1 Ambos binomios son una suma
(2x)(2x) + (2x)(3) + (5)(2x) + (5)(3) =
4x2
+ 6x + 10x + [(5)(3)] =
4x2
+ (6 + 10)x + [(5)(3)] = 4x2 + 16x + 15
(a + c)(a + b) = a2 + (b + c)a + [(c)(b)]
(a)(a) + (a)(b) + (c)(a) + (c)(b) =
a2
+ a(b + c) + [(c)(b)] = a2
+ (b + c)a + [(c)(b)]
(4x – 2)(4x – 7) = CASO 2 Ambos binomios son una resta
(4x)(4x) + (4x)(– 7) + (– 2)(4x) + (– 2)(– 7) =
16x2
– 28x – 8x + [(– 2)(– 7)] =
4x2
– (28 + 8)x + [(– 2)(– 7)] = 4x2 – 36x + 14
(a – c)(a – b) = a2 – (b + c)a + [(b)(c)]
(a)(a) + a(– b) + (– c)(a) + (–c)(– b) =
a2
+ (–b – c)a + [(–b)(– c)] = a2
– (b + c)a + [(b)(c)]
(3x + 9)(5x – 4) = CASO 3 Un binomio suma y otro resta
(3x)(5x) + (3x)(– 4) + (9)(5x) + (9)(– 4) =
15x2
– 12x + 45x + [(9)(– 4)] =
4x2
+ (–12 + 45)x – [(9)(4)] = 4x2 + 33x – 36
(a + c)(a – b) = a2 + (c – b)a – [(b)(c)]
(a)(a) + a(– b) + c(a) + (c)(– b) =
a2
+ (c – b)a + [(–b)(c)] = a2
+ (c – b)a – [(b)(c)]
Cubo de la suma de un binomio
(5x + 1)3
=
(5x + 1)(5x + 1)(5x + 1) =
(25x2
+ 5x + 5x + 1)(5x + 1) =

3. (25x2
+ 10x + 1)(5x + 1) =
125x3
+ 50x2
+ 5x + 25x2
+ 10x + 1=
125x3
+ 75x2
+ 15x + 1
(a + b)3
= a3 + 3a2b + 3ab2 + b3
(a + b)(a + b)(a + b) =
(a2
+ ab + ab + b2
)(a + b) =
(a2
+ 2ab + b2
)(a + b) =
a3
+ 2a2
b + ab2
+ a2
b + 2ab2
+ b3
= a3
+ 3a2
b + 3ab2
+ b3
Cubo de la diferencia de un binomio
(3x – 5)3
=
(3x – 5)(3x – 5)(3x – 5) =
(9x2
– 15x – 15x + 25)(3x – 5) =
(9x2
– 30x + 25)(3x – 5) =
27x3
– 90x2
+ 75x – 45x2
+ 150x – 125 =
27x3
– 135x2
+ 225x – 125
(a – b)3
= a3 – 3a2b + 3ab2 – b3
(a – b)(a – b)(a – b) =
(a2
– ab – ab + b2
)(a – b) =
(a2
– 2ab + b2
)(a – b) =
a3
– 2a2
b + ab2
– a2
b + 2ab2
– b3
= a3
– 3a2
b + 3ab2
– b3