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Study Guides > College Algebra

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{x}^{2}+5x+6 has a GCF of 1, but it can be written as the product of the factors (x+2)\left(x+2\right) and (x+3)\left(x+3\right). Trinomials of the form x2+bx+c{x}^{2}+bx+c can be factored by finding two numbers with a product of cc and a sum of bb. The trinomial x2+10x+16{x}^{2}+10x+16, for example, can be factored using the numbers 22 and 88 because the product of those numbers is 1616 and their sum is 1010. The trinomial can be rewritten as the product of (x+2)\left(x+2\right) and (x+8)\left(x+8\right).

A General Note: Factoring a Trinomial with Leading Coefficient 1

A trinomial of the form x2+bx+c{x}^{2}+bx+c can be written in factored form as (x+p)(x+q)\left(x+p\right)\left(x+q\right) where pq=cpq=c and p+q=bp+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{x}^{2}+bx+c, factor it.

  1. List factors of cc.
  2. Find pp and qq, a pair of factors of cc with a sum of bb.
  3. Write the factored expression (x+p)(x+q)\left(x+p\right)\left(x+q\right).

Example 2: Factoring a Trinomial with Leading Coefficient 1

Factor x2+2x−15{x}^{2}+2x - 15.

Solution

We have a trinomial with leading coefficient 1,b=21,b=2, and c=−15c=-15. We need to find two numbers with a product of −15-15 and a sum of 22. In the table, we list factors until we find a pair with the desired sum.
 
Factors of −15-15 Sum of Factors
1,−151,-15 −14-14
−1,15-1,15 14
3,−53,-5 −2-2
−3,5-3,5 2
Now that we have identified pp and qq as −3-3 and 55, write the factored form as (x−3)(x+5)\left(x - 3\right)\left(x+5\right).

Analysis of the Solution

We can check our work by multiplying. Use FOIL to confirm that (x−3)(x+5)=x2+2x−15\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 x2−7x+6{x}^{2}-7x+6. Solution

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