7.3 Factor Trinomials of the Form ax2+bx+c

Learning Objectives

By the end of this section, you will be able to:

  • Recognize a preliminary strategy to factor polynomials completely
  • Factor trinomials of the form ax2+bx+c with a GCF
  • Factor trinomials using trial and error
  • Factor trinomials using the ‘ac’ method

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, x2+bx+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.

How To

Choose a strategy to factor polynomials completely.

  1. Step 1. Is there a greatest common factor?
    • Factor it out.
  2. Step 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 x2+bx+c

: Undo FOIL (x)(x)

  • If it has more than three terms: Use the grouping method.

Step 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.

Factor Trinomials of the form ax2 + bx + c with a GCF

Now that we have organized what we’ve covered so far, we are ready to factor trinomials whose leading coefficient is not 1, trinomials of the form ax2+bx+c

Remember to always check for a GCF first! Sometimes, after you factor the GCF, the leading coefficient of the trinomial becomes 1 and you can factor it by the methods in the last section. Let’s do a few examples to see how this works.

Watch out for the signs in the next two examples.

In the next example the GCF will include a variable.

Factor Trinomials using Trial and Error

What happens when the leading coefficient is not 1 and there is no GCF? There are several methods that can be used to factor these trinomials. First we will use the Trial and Error method.

Let’s factor the trinomial 3x2+5x+2.

From our earlier work we expect this will factor into two binomials.

We know the first terms of the binomial factors will multiply to give us 3x2. The only factors of 3x 2 are 1x,3x. We can place them in the binomial.

Check. Does 1x⋅3x=3x2?

We know the last terms of the binomials will multiply to 2. Since this trinomial has all positive terms, we only need to consider positive factors. The only factors of 2 are 1 and 2. But we now have two cases to consider as it will make a difference if we write 1, 2, or 2, 1.

This figure demonstrates the possible factors of the polynomial 3x^2 +5x +2. The polynomial is written twice. Underneath both, there are the terms 1x, 3x under the 3x^2. Also, there are the factors 1,2 under the 2 term. At the bottom of the figure there are two possible factorizations of the polynomial. The first is (x + 1)(3x + 2) and the next is (x + 2)(3x + 1).

Which factors are correct? To decide that, we multiply the inner and outer terms.

This figure demonstrates the possible factors of the polynomial 3 x^ 2 + 5 x +2. The polynomial is written twice. Underneath both, there are the terms 1 x, 3 x under the 3 x ^ 2. Also, there are the factors 1, 2 under the 2 term. At the bottom of the figure there are two possible factorizations of the polynomial. The first is (x + 1)(3 x + 2). Underneath this factorization are the products 3 x from multiplying the middle terms 1 and 3 x. Also there is the product of 2 x from multiplying the outer terms x and 2. These products of 3 x and 2 x add to 5 x. Underneath the second factorization are the products 6 x from multiplying the middle terms 2 and 3 x. Also there is the product of 1 x from multiplying the outer terms x and 1. These two products of 6 x and 1 x add to 7 x.

Since the middle term of the trinomial is 5x, the factors in the first case will work. Let’s FOIL to check.

Our result of the factoring is:

How To

Factor trinomials of the form ax2+bx+c

using trial and error.

  • Step 1. Write the trinomial in descending order of degrees.
  • Step 2. Find all the factor pairs of the first term.
  • Step 3. Find all the factor pairs of the third term.
  • Step 4. Test all the possible combinations of the factors until the correct product is found.
  • Step 5. Check by multiplying.

When the middle term is negative and the last term is positive, the signs in the binomials must both be negative.

When we factor an expression, we always look for a greatest common factor first. If the expression does not have a greatest common factor, there cannot be one in its factors either. This may help us eliminate some of the possible factor combinations.

Don’t forget to look for a GCF first.

Factor Trinomials using the “ac” Method

Another way to factor trinomials of the form ax2+bx+c is the “ac” method. (The “ac” method is sometimes called the grouping method.) The “ac” method is actually an extension of the methods you used in the last section to factor trinomials with leading coefficient one. This method is very structured (that is step-by-step), and it always works!

When the third term of the trinomial is negative, the factors of the third term will have opposite signs.

Don’t forget to look for a common factor!

We can now update the Preliminary Factoring Strategy, as shown in Figure 7.2 and detailed in Choose a strategy to factor polynomials completely (updated), to include trinomials of the form ax2+bx+c. Remember, some polynomials are prime and so they cannot be factored.

This figure has the strategy for factoring polynomials. At the top of the figure is GCF. Below this, there are three options. The first is binomial. The second is trinomial. Under trinomial there are x squared + b x + c and a x squared + b x +c. The two methods here are trial and error and the “a c” method. The third option is for more than three terms. It is grouping.
This figure has the strategy for factoring polynomials. At the top of the figure is GCF. Below this, there are three options. The first is binomial. The second is trinomial. Under trinomial there are x squared + b x + c and a x squared + b x +c. The two methods here are trial and error and the “a c” method. The third option is for more than three terms. It is grouping.
Figure 7.2

Media

Access these online resources for additional instruction and practice with factoring trinomials of the form ax2+bx+c.

Section 7.3 Exercises

Practice Makes Perfect

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

Mixed Practice

In the following exercises, factor.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.This table has the following statements all to be preceded by “I can…”. The first row is “recognize a preliminary strategy to factor polynomials completely”. The second row is “factor trinomials of the form a x ^ 2 + b x + c with a GCF”. The third row is “factor trinomials using trial and error”. And the fourth row is “factor trinomials using the “ac” method”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?