Factoring Quadratic Polynomials

Algebraic Transformations

Algebraic Expressions

Polynomials

Algebraic Transformations

Outline

Factoring Quadratic Polynomials

Other Transformations

Expanding

Factoring Quadratic Polynomials

Problem 1. Factor the difference between two squares, a2 – b2.
Solution. First, subtract and add the product ab:

a2 – b2 = a2 – a b + a b – b2.

Then combine the terms by pairs and take out the common factors:

a2 – a b + a b – b2 = (a2 – a b) + (a b – b2)
= a (a – b) + b (a – b) = (a – b) (a + b).

Therefore,

a2 – b2 = (a – b) (a + b)

Problem 2. Factor the quadratic trinomial, a2 + 2 a b + b2.

Solution. First, rewrite the term 2 a b as a b + a b.
Then combine the terms by pairs and take out the common factors:

a2 + 2 a b + b2 = a2 + a b + a b + b2
= (a2 + a b) + (a b + b2)
= a(a + b) + b (a + b) = (a + b)(a + b).

Therefore, we get the following formula for the perfect square trinomial:

a2 + 2 a b + b2 = (a + b)2

Corollary. Substituting (–b) for b, we obtain the following formula for the difference squared:

a2 – 2 a b + b2 = (a – b)2