Topic: Factoring
Subtopic: Special Forms & General Strategy
Overview
This last section on factoring polynomial expressions requires using three factoring formulas: difference of squares, difference of cubes, and sum of cubes.
- Caution: a binomial squared like (x+3)^2 must be FOILed out to get the "perfect square trinomial" x^2+6x+9 - never take the power across the addition to get x^2+3^2! Keeping this in mind will help you recognize PSTs when working backwards to factor into a binomial squared.
- If the original poly has only two terms then it factors as either a difference of squares, difference of cubes, or sum of cubes. These formulas are worth memorizing.
- Sum of squares are PRIME! E.g. x^2+9 cannot be factored (well, not until we cover imaginary numbers).
- Factoring polynomials enables us to solve polynomial equations via the zero product rule. The "zero products rule" is only applicable to equations not expressions. That is, you cannot "solve" the expression x^2+6x+9, all you can do is factor it.
Objectives
By the end of this topic you should know and be prepared to be tested on:
- 7.3.1 Recognize perfect square trinomials and be able to factor them down as a binomial squared
- 7.3.2 Recognize difference of squares and be able to factor them down including repeated difference of squares
- 7.3.3 Know that sum of squares can't be factored assuming any GCF is already factored out
- 7.3.4 Recognize sum/difference of cubes and be able to factor them down according to the formulas
Terminology
Define: binomial squared, PST = perfect square trinomial, difference of squares, sum/difference of cubes
Text Notes
Text:
Intro & Inter Algebra for CS 3ed by Blitzer, sect. 6.4-6.5
- ch 6.4 includes factoring special forms: difference of squares that factor into the form (_+_)(_-_), perfect square trinomials that factor into a binomial square ( )2, and sum or difference of cubes - the formulas on pg 430 are worth putting in your notes.
- ch 6.4 You may skip "factoring geometrically" if you want.
- ch 6.5 It is very important to practice some exercises in this section since they combine all the factoring methods and mix the processes so you must determine which method of factoring is to be used when and in which order.
- pg 456 "Section 6.5 General Factoring Strategy" provides a useful overview of the factoring strategies.