Factoring a Trinomial with Leading Coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial ${x}^{2}+5x+6$ has a GCF of 1, but it can be written as the product of the factors $\left(x+2\right)$ and $\left(x+3\right)$ .

Trinomials of the form ${x}^{2}+bx+c$ can be factored by finding two numbers with a product of $c$ and a sum of $b$ . The trinomial ${x}^{2}+10x+16$ , for example, can be factored using the numbers $2$ and $8$ because the product of those numbers is $16$ and their sum is $10$ . The trinomial can be rewritten as the product of $\left(x+2\right)$ and $\left(x+8\right)$ .

A General Note: Factoring a Trinomial with Leading Coefficient 1

A trinomial of the form ${x}^{2}+bx+c$ can be written in factored form as $\left(x+p\right)\left(x+q\right)$ where $pq=c$ and $p+q=b$ .

Q & A

Can every trinomial be factored as a product of binomials?

No. Some polynomials cannot be factored. These polynomials are said to be prime.

How To: Given a trinomial in the form ${x}^{2}+bx+c$ , 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 $\left(x+p\right)\left(x+q\right)$ .

Example 2: Factoring a Trinomial with Leading Coefficient 1

Factor ${x}^{2}+2x - 15$ .

Solution

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 $\left(x - 3\right)\left(x+5\right)$ .

Analysis of the Solution

We can check our work by multiplying. Use FOIL to confirm that $\left(x - 3\right)\left(x+5\right)={x}^{2}+2x - 15$ .

Q & A

Does the order of the factors matter?

No. Multiplication is commutative, so the order of the factors does not matter.

Try It 2

Factor ${x}^{2}-7x+6$ .

Solution

Factoring Polynomials