Steps for Factoring Polynomials

Sometimes I meet students in Math 65, 70, or 95 who were absent the day of a very important lesson for success in those classes: how to know which of the six previously studied factoring methods to use when the homework or test simply gives you a polynomial to factor.

(This overview used to be on page 427 of the Math 70 textbook. I'm not sure if a new edition has changed this.)

Prepare

Make sure the terms are in order.

Bring to the front any Greatest Common Factor.

Toolbox

A student in Math 65, 70, or 95 has only been taught a "toolbox" of six "tools" for factoring. Any homework problem will always fit one of these tools because the textbook is not going to ask you to do a problem that you have not been taught how to solve.

The available tools can be organized by how many terms the problem has. You will only need to check at most three tools before identifying which tool is appropriate for the problem.

Two Terms

For a polynomial with two terms, you have two tools at your disposal. Either the format is A² − B² or A³ ± B³.

Three Terms

For a polynomial with three terms, you have three tools at your disposal. First check if the format is (A ± B)². If it is not, then use either guess-and-check or factor-by-grouping.

Four Terms

For a polynomial with four terms, you have only one tool at your disposal. The format will always be an incomplete factor-by-grouping problem which you need to finish.

Finish

Check that all polynomials with variables are prime.

If you have time, check your answer by multiplying.