CHAPTER 8 Polynomials
8.5 Factor Quadratic Trinomials with Leading Coefficient 1
Learning Objectives
By the end of this section, you will be able to:
- Factor trinomials of the form
- Factor trinomials of the form
Factor Trinomials of the Form x2 + bx + c
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, , and end with the factors, . 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 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 the table below—the table will be very helpful when you work with numbers that can be factored in many different ways.
| Factors of | Sum of factors |
|---|---|
We see that 2 and 3 are the numbers that multiply to 6 and add to 5. So we have the factors of . They are .
You should check this by multiplying.
Looking back, we started with , which is of the form , where and . We factored it into two binomials of the form and .
To get the correct factors, we found two numbers m and n whose product is c and sum is b.
EXAMPLE 1
Factor: .
TRY IT 1.1
Factor: .
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TRY IT 1.2
Factor: .
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Let’s summarize the steps we used to find the factors.
HOW TO:
Factor trinomials of the form .
- Write the factors as two binomials with first terms x: .
- Find two numbers m and n that
Multiply to c,
Add to b, - Use m and n as the last terms of the factors: .
- Check by multiplying the factors.
EXAMPLE 2
Factor: .
Notice that the variable is u, so the factors will have first terms u.
Find two numbers that: multiply to 24 and add to 11
| Factors of | Sum of factors |
|---|---|
Use 3 and 8 as the last terms of the binomials.
Check.
TRY IT 2.1
Factor: .
Show answer
TRY IT 2.2
Factor: .
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EXAMPLE 3
Factor: .
Find two numbers that multiply to 60 and add to 17
| Factors of | Sum of factors |
|---|---|
Use 5 and 12 as the last terms.
Check.
TRY IT 3.1
Factor: .
Show answer
TRY IT 3.2
Factor: .
Show answer
Factor Trinomials of the Form x2 + bx + c with b Negative, c Positive
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 numbe