## 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 $p+q=b$.

### 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 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 $\left(x - 3\right)\left(x+5\right)={x}^{2}+2x - 15$.

### 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