Factoring

Some polynomials have special forms. In this case we have formulas we can
use to do the factoring

If a problem cannot be factored then we say that it is prime.

Definition 0.1 Difference of Squares
If A and B are real numbers , variables , or algebraic expressions, then

Example 0.1 Difference of Square

where A = x and B = 4.

where A = 2x and B = 3.

where A = xy and B = 10.

where A = 5x² and B = 3y³.

where A = x - 6 and B = y.

Sometimes after an expression is factored , more factoring will need to be
done. Always check that the expression cannot be factored any further.

Example 0.2 Repeated Factoring

where A = x² and B = 1.

But notice one of the factors is again a difference of squares .

Definition 0.2 Sum/Difference of Cubes

1. Factoring the Sum of Cubes

2. Factoring the Difference of Cubes

To remember the signs recall SOAP. Stands for same, opposite, always
positive .

First signs are same, next is opposite the first, last sign is always positive .

Example 0.3 Sum/Difference of Cubes

where A = x and B = 1

where A = x and B = 10

where A = 3x and B = 2

where A = xy and B = 4

where A = 6 and B = x

so

Property 0.1 Factoring Perfect Square Polynomials

Trinomials obtained from squaring a binomial are called perfect square tri-
nomials and have a special form.

so

and

so

Example 0.4 Perfect Square Trinomial

If this is a perfect square trinomial then A = x and B = 1 making
2AB = 2(x)(1) = 2x which it is.
So this is a perfect square trinomial.

If this is a perfect square trinomial then A = x and B = 7 making
2AB = 2(x)(7) = 14x which it is.
So this is a perfect square trinomial.

If this is a perfect square trinomial then A = 5y and B = 1 making
2AB = 2(5y)(1) = 10y which it is.
So this is a perfect square trinomial.

If this is a perfect square trinomial then A = 3x and B = 8y making
2AB = 2(3x)(8y) = 48xy which it is.
So this is a perfect square trinomial.