## 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 ${x}^{2}+5x+6$ has a GCF of $1$, but it can be written as the product of the factors $\left(x+2\right)$ and $\left(x+3\right)$.

Recall how to use the distributive property to multiply two binomials:

$\left(x+2\right)\left(x+3\right) = x^2+3x+2x+6=x^2+5x+6$

We can reverse the distributive property and return $x^2+5x+6\text{ to }\left(x+2\right)\left(x+3\right)$ 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 ${x}^{2}+bx+c$, 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 ${x}^{2}+2x - 15$.

In the following video, we present two more examples of factoring a trinomial with a leading coefficient of 1.

To summarize our process, consider the following steps:

### 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 $\left(x+p\right)\left(x+q\right)$.

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 $x^{2}+x–12$.

Which property of multiplication can be used to describe why $\left(x+4\right)\left(x-3\right) =\left(x-3\right)\left(x+4\right)$. Use the textbox below to write down your ideas before you look at the answer.

In our last example, we will show how to factor a trinomial whose b term is negative.

### Example

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

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.

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?