By the end of this section, you will be able to:
You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication—to start with the product and end up with the factors. Let’s look at an example of multiplying binomials to refresh your memory.
To factor the trinomial means to start with the product, x2+5x+6, and end with the factors, (x+2)(x+3). You need to think about where each of the terms in the trinomial came from.
The first term came from multiplying the first term in each binomial. So to get x2 in the product, each binomial must start with an x.
The last term in the trinomial came from multiplying the last term in each binomial. So the last terms must multiply to 6.
What two numbers multiply to 6?
The factors of 6 could be 1 and 6, or 2 and 3. How do you know which pair to use?
Consider the middle term. It came from adding the outer and inner terms.
So the numbers that must have a product of 6 will need a sum of 5. We’ll test both possibilities and summarize the results in Table 7.1—the table will be very helpful when you work with numbers that can be factored in many different ways.
We see that 2 and 3 are the numbers that multiply to 6 and add to 5. So we have the factors of x2+5x+6. They are (x+2)(x+3).
You should check this by multiplying.
Looking back, we started with x2+5x+6, which is of the form x2+bx+c, where b=5 and c=6. We factored it into two binomials of the form (x+m)and(x+n).
Let’s summarize the steps we used to find the factors.
In the examples so far, all terms in the trinomial were positive. What happens when there are negative terms? Well, it depends which term is negative. Let’s look first at trinomials with only the middle term negative.
Remember: To get a negative sum and a positive product, the numbers must both be negative.
Again, think about FOIL and where each term in the trinomial came from. Just as before,
How do you get a positive product and a negative sum? With two negative numbers.
Now, what if the last term in the trinomial is negative? Think about FOIL. The last term is the product of the last terms in the two binomials. A negative product results from multiplying two numbers with opposite signs. You have to be very careful to choose factors to make sure you get the correct sign for the middle term, too.
Remember: To get a negative product, the numbers must have different signs.
Let’s make a minor change to the last trinomial and see what effect it has on the factors.
Some trinomials are prime. The only way to be certain a trinomial is prime is to list all the possibilities and show that none of them work.
Let’s summarize the method we just developed to factor trinomials of the form x2+bx+c.
Sometimes you’ll need to factor trinomials of the form x2+bxy+cy2 with two variables, such as x 2 +12xy+36y 2 . The first term, x 2 , is the product of the first terms of the binomial factors, x⋅x. The y 2 in the last term means that the second terms of the binomial factors must each contain y. To get the coefficients b and c, you use the same process summarized in the previous objective.
Factor Trinomials of the Form x2+bx+c
In the following exercises, factor each trinomial of the form x2+bx+c.
129.Consecutive integers Deirdre is thinking of two consecutive integers whose product is 56. The trinomial x2+x−56 describes how these numbers are related. Factor the trinomial.
130.Consecutive integers Deshawn is thinking of two consecutive integers whose product is 182. The trinomial x2+x−182 describes how these numbers are related. Factor the trinomial.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all goals?