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7.3 Factor quadratic trinomials with leading coefficient other than

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By the end of this section, you will be able to:
  • Recognize a preliminary strategy to factor polynomials completely
  • Factor trinomials of the form a x 2 + b x + c with a GCF
  • Factor trinomials using trial and error
  • Factor trinomials using the ‘ac’ method

Before you get started, take this readiness quiz.

  1. Find the GCF of 45 p 2 and 30 p 6 .
    If you missed this problem, review [link] .
  2. Multiply ( 3 y + 4 ) ( 2 y + 5 ) .
    If you missed this problem, review [link] .
  3. Combine like terms 12 x 2 + 3 x + 5 x + 9 .
    If you missed this problem, review [link] .

Recognize a preliminary strategy for factoring

Let’s summarize where we are so far with factoring polynomials. In the first two sections of this chapter, we used three methods of factoring: factoring the GCF, factoring by grouping, and factoring a trinomial by “undoing” FOIL. More methods will follow as you continue in this chapter, as well as later in your studies of algebra.

How will you know when to use each factoring method? As you learn more methods of factoring, how will you know when to apply each method and not get them confused? It will help to organize the factoring methods into a strategy that can guide you to use the correct method.

As you start to factor a polynomial, always ask first, “Is there a greatest common factor?” If there is, factor it first.

The next thing to consider is the type of polynomial. How many terms does it have? Is it a binomial? A trinomial? Or does it have more than three terms?

If it is a trinomial where the leading coefficient is one, x 2 + b x + c , use the “undo FOIL” method.

If it has more than three terms, try the grouping method. This is the only method to use for polynomials of more than three terms.

Some polynomials cannot be factored. They are called “prime.”

Below we summarize the methods we have so far. These are detailed in Choose a strategy to factor polynomials completely .

This figure lists strategies for factoring polynomials. At the top of the figure is G C F, where factoring always starts. From there, the figure has three branches. The first is binomial, the second is trinomial with the form x ^ 2 + b x +c, and the third is “more than three terms”, which is labeled with grouping.

Choose a strategy to factor polynomials completely.

  1. Is there a greatest common factor?
    • Factor it out.
  2. Is the polynomial a binomial, trinomial, or are there more than three terms?
    • If it is a binomial, right now we have no method to factor it.
    • If it is a trinomial of the form x 2 + b x + c : Undo FOIL ( x ) ( x )
    • If it has more than three terms: Use the grouping method.
  3. Check by multiplying the factors.

Use the preliminary strategy to completely factor a polynomial. A polynomial is factored completely if, other than monomials, all of its factors are prime.

Identify the best method to use to factor each polynomial.

  1. ⓐ 6 y 2 − 72
  2. ⓑ r 2 − 10 r − 24
  3. ⓒ p 2 + 5 p + p q + 5 q

Solution

  1. ⓐ
    6 y 2 − 72 Is there a greatest common factor? Yes, 6. Factor out the 6. 6 ( y 2 − 12 ) Is it a binomial, trinomial, or are there Binomial, we have no method to factor more than 3 terms? binomials yet.

  2. ⓑ
    r 2 − 10 r − 24 Is there a greatest common factor? No, there is no common factor. Is it a binomial, trinomial, or are there Trinomial, with leading coefficient 1, so more than three terms? “undo” FOIL.

  3. ⓒ
    p 2 + 5 p + p q + 5 q Is there a greatest common factor? No, there is no common factor. Is it a binomial, trinomial, or are there More than three terms, so factor using more than three terms? grouping.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Identify the best method to use to factor each polynomial:

  1. ⓐ 4 y 2 + 32
  2. ⓑ y 2 + 10 y + 21
  3. ⓒ y z + 2 y + 3 z + 6

ⓐ no method ⓑ undo using FOIL ⓒ factor with grouping

Got questions? Get instant answers now!

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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