# 4.3 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 $\left(x+8\right)$.

## 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 $p+q=b$.

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.

1. List factors of $c$.
2. Find $p$ and $q$, a pair of factors of $c$ with a sum of $b$.
3. Write the factored expression $\left(x+p\right)\left(x+q\right)$.

## Example 1: Factoring a Trinomial with Leading Coefficient 1

1. Factor $x^2+2x−15$.
2. Factor $x^2−7x+6$.

### Analysis

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