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 x2+5x+6 has a GCF of 1, but it can be written as the product of the factors (x+2) and (x+3).

Trinomials of the form x2+bx+c can be factored by finding two numbers with a product of c and a sum of b. The trinomial x2+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 (x+2) and (x+8).

A General Note: Factoring a Trinomial with Leading Coefficient 1

A trinomial of the form x2+bx+c can be written in factored form as (x+p)(x+q) 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 x2+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 (x+p)(x+q).

Example 2: Factoring a Trinomial with Leading Coefficient 1

Factor x2+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 (x−3)(x+5).

Analysis of the Solution

We can check our work by multiplying. Use FOIL to confirm that (x−3)(x+5)=x2+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 x2−7x+6.

Solution