By the end of this section, you will be able to:
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.
: Undo FOIL (x)(x)
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.
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.
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.
Which factors are correct? To decide that, we multiply the inner and outer terms.
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:
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.
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.
Access these online resources for additional instruction and practice with factoring trinomials of the form ax2+bx+c.
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.
In the following exercises, factor.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?