Factor a Trinomial with Leading Coefficient = 1
Learning Outcomes
- Factor a trinomial with leading coefficient =1
Trinomials are polynomials with three terms. We are going to show you a method for factoring a trinomial whose leading coefficient is 1. Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that trinomials can be factored. The trinomial x2+5x+6 has a GCF of 1, but it can be written as the product of the factors (x+2) and (x+3).
Recall how to use the distributive property to multiply two binomials:
(x+2)(x+3)=x2+3x+2x+6=x2+5x+6
We can reverse the distributive property and return x2+5x+6 to (x+2)(x+3) by finding two numbers with a product of 6 and a sum of 5.
Factoring a Trinomial with Leading Coefficient 1
In general, for a trinomial of the form
x2+bx+c, you can factor a trinomial with leading coefficient
1 by finding two numbers,
p and
q whose product is c and whose sum is b.
Let us put this idea to practice with the following example.
Example
Factor
x2+2x−15.
Answer:
We have a trinomial with leading coefficient 1,b=2, and c=−15. We need to find two numbers with a product of −15 and a sum of 2. In the table, we list factors until we find a pair with the desired sum.
Factors of −15 |
Sum of Factors |
1,−15 |
−14 |
−1,15 |
14 |
3,−5 |
−2 |
−3,5 |
2 |
Now that we have identified
p and
q as
−3 and
5, write the factored form as
(x−3)(x+5).
In the following video, we present two more examples of factoring a trinomial with a leading coefficient of 1.
https://youtu.be/-SVBVVYVNTM
To summarize our process, consider the following steps:
How To: Given a trinomial in the form x2+bx+c, factor it
- List factors of c.
- Find p and q, a pair of factors of c with a sum of b.
- Write the factored expression (x+p)(x+q).
We will now show an example where the trinomial has a negative c term. Pay attention to the signs of the numbers that are considered for p and q.
In our next example, we show that when c is negative, either p or q will be negative.
Example
Factor
x2+x–12.
Answer:
Consider all the combinations of numbers whose product is −12 and list their sum.
Factors whose product is −12 |
Sum of the factors |
1⋅−12=−12 |
1+−12=−11 |
2⋅−6=−12 |
2+−6=−4 |
3⋅−4=−12 |
3+−4=−1 |
4⋅−3=−12 |
4+−3=1 |
6⋅−2=−12 |
6+−2=4 |
12⋅−1=−12 |
12+−1=11 |
Choose the values whose sum is
+1:
p=4 and
q=−3, and place them into a product of binomials.
(x+4)(x−3)
Think About It
Which property of multiplication can be used to describe why
(x+4)(x−3)=(x−3)(x+4). Use the textbox below to write down your ideas before you look at the answer.
[practice-area rows="2"][/practice-area]
Answer:
The commutative property of multiplication states that factors may be multiplied in any order without affecting the product.
a⋅b=b⋅a
In our last example, we will show how to factor a trinomial whose b term is negative.
Example
Factor
x2−7x+6.
Answer:
List the factors of 6. Note that the b term is negative, so we will need to consider negative numbers in our list.
Factors of 6 |
Sum of Factors |
1,6 |
7 |
2,3 |
5 |
−1,−6 |
−7 |
−2,−3 |
−5 |
Choose the pair that sum to
−7, which is
−1,−6
Write the pair as constant terms in a product of binomials.
(x−1)(x−6)
In the last example, the b term was negative and the c term was positive. This will always mean that if it can be factored, p and q will both be negative.
Think About It
Can every trinomial be factored as a product of binomials?
Mathematicians often use a counterexample to prove or disprove a question. A counterexample means you provide an example where a proposed rule or definition is not true. Can you create a trinomial with leading coefficient
1 that
cannot be factored as a product of binomials?
Use the textbox below to write your ideas.
[practice-area rows="2"][/practice-area]
Answer:
Can every trinomial be factored as a product of binomials? No. Some polynomials cannot be factored. These polynomials are said to be prime.
A counterexample would be: x2+3x+7
Licenses & Attributions
CC licensed content, Original
- Factor a Trinomial Using the Shortcut Method - Form x^2+bx+c. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Unit 12: Factoring, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.