6.6: Summary of Factoring

section 6.6 Learning Objectives

6.6: Summary of Factoring

  • Review the factoring methods presented in this module
  • Be able to recognize and apply an appropriate factoring technique to a given problem
  • Factor expressions completely

 

In the previous sections, you have learned several factoring techniques. Now it is time to put it all together.

In this section, we focus on two important questions you should ask yourself when encountering any factoring problem:

  1. Which factoring technique should I use for this problem?
  2. Can I factor the polynomial more?

Choosing the Best Factoring Technique

Here, we present a strategy you can apply to any factoring problem.

Factoring strategy

1. If there is a GCF other than [latex]1[/latex], factor it out first. Don’t forget to factor out a [latex]-1[/latex] if the leading coefficient is negative.

2. Count the number of terms in the remaining polynomial and select an appropriate technique.

I.  4 or More Terms:  Factor by Grouping (Section 6.2)

II.  3 Terms:

A. If [latex]a=1[/latex], apply the “Product and Sum Method” (Section 6.3)

B. If [latex]a\neq 1[/latex], apply the “AC-Method” (Section 6.4)

C. If it is a perfect square trinomial, use the appropriate formula: [latex]a^2+2ab+b^2=(a+b)^2[/latex] or [latex]a^2-2ab+b^2=(a-b)^2[/latex] (Section 6.5)

III.  2 Terms: If the binomial is a difference of squares, use the following formula: [latex]a^2-b^2=(a+b)(a-b)[/latex] (Section 6.5)

3. If none of the above applies, it is possible that the polynomial is not factorable, or “prime.”

Let us try some examples.

Example 1

Factor [latex]-3x^2-3x+6[/latex].

Example 2

Factor [latex]-3x^2-7x+6[/latex].

The next example includes a perfect square trinomial.

Example 3

Factor [latex]12x^5+60x^4+75x^3[/latex]

 

In the next two examples, we review factoring binomials.

Example 4

Factor [latex]-2x^2+8[/latex]

 

Example 5

Factor [latex]50x^{2}y^{3}-8y[/latex]

Don’t forget that we cannot factor every polynomial.

Example 6

Factor [latex]5x^2+8x+4[/latex]

 

Factoring More

Sometimes after we factor, one or more of the resulting polynomials can be factored even further. You will see this more in future classes, but present one example here for you to think about.

think about it

Factor completely: [latex]x^4-16[/latex]

 

Now that you have mastered several factoring techniques, we finally explore some applications of factoring in the next section.