Special Products of Polynomials

Key Questions

• The general form for multiplying two binomials is:
$\left(x + a\right) \left(x + b\right) = {x}^{2} + \left(a + b\right) x + a b$

Special products:

1. the two numbers are equal, so it's a square:
$\left(x + a\right) \left(x + a\right) = {\left(x + a\right)}^{2} = {x}^{2} + 2 a x + {a}^{2}$, or
$\left(x - a\right) \left(x - a\right) = {\left(x - a\right)}^{2} = {x}^{2} - 2 a x + {a}^{2}$
Example : ${\left(x + 1\right)}^{2} = {x}^{2} + 2 x + 1$
Or: ${51}^{2} = {\left(50 + 1\right)}^{2} = {50}^{2} + 2 \cdot 50 + 1 = 2601$

2. the two numbers are equal, and opposite sign:
$\left(x + a\right) \left(x - a\right) = {x}^{2} - {a}^{2}$
Example : $\left(x + 1\right) \left(x - 1\right) = {x}^{2} - 1$
Or: $51 \cdot 49 = \left(50 + 1\right) \left(50 - 1\right) = {50}^{2} - 1 = 2499$

A trinomial that when factored gives you the square of a binomial

Explanation:

Given: What is a perfect square binomial?

A perfect square binomial is a trinomial that when factored gives you the square of a binomial.

Ex. ${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$

Ex. ${\left(2 a + 3 b\right)}^{2} = 4 {a}^{2} + 12 a b + 9 {b}^{2}$