Chapter 8 - Factoring

Before you begin to learn about factoring expressions you may want to review factoring numbers, because factoring expressions is based on it. Also you previously learned how to multiply expressions. Factoring is the reverse process, in that you find those quantities that would be multiplied to give some expression. So you may want to review that topic as well.

Before learning about factoring you might want to know what it's good for. Factoring is required for:

Simplifying expressions; for example combining like terms.
Solving equations; for example quadratic equations.
Simplifying fractions and simplifying algebraic fractions.
Adding common fractions and adding algebraic fractions.
Completing the square.

Some definitions:

The factors of an expression are those quantities whose product yields the expression.
Factoring is the process of finding the factors of an expression.
An expression is said to be prime if it has no factors other than 1 and itself.
An expression that is not prime is said to be factorable.

Example: The prime factors of the monomial 5 a x² are obviously 5, a, x and x. Factoring monomials is so simple we discuss them no further. From now on we will assume that we want to factor multinomials.

Example: The factors of the expression a x + b x are x and a + b, because the expression can be written as the product of the factors, like this:

a x + b x = x (a + b)

You can verify that this is correct by multiplying out the factored expression on the right side and obtaining the original expression on the left side.

Example: The factors of x² − 4 are x − 2 and x + 2, because:

Again, you can verify that this is correct by multiplying out the factored expression on the right side and obtaining the original expression on the left side.

Example: The expressions x − 2 and x + 2 are prime, but as the previous example shows, the expression x² − 4 is factorable.

Factoring an expression is done by recognizing the form of the expression and then applying the rules for factoring that form. The following table gives the names of the factorable forms and an example of each. The left side of the example shows the expression before factoring and the right side shows the expression after factoring. The forms are ordered more or less from the simplest to the hardest to factor. This is the order in which the Algebra Coach tests an expression for factorability. Click on any form to see how it is factored.

Factorable form or method used Example

Common factor

Difference of squares

Quadratic trinomial, perfect square

Quadratic trinomial, a = 1

Quadratic trinomial, a ≠ 1

Completing the square

Grouping

Sum or difference of cubes

Polynomial

Factorable form or method used	Example
Common factor
Difference of squares
Quadratic trinomial, perfect square
Quadratic trinomial, a = 1
Quadratic trinomial, a ≠ 1
Completing the square
Grouping
Sum or difference of cubes
Polynomial