Factoring Basics

Learning Objectives

  • Identify and factor the GCF of a polynomial
  • Factor a Trinomial with Leading Coefficient 1
  • Factor by Grouping


When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, [latex]4[/latex] is the GCF of [latex]16[/latex] and [latex]20[/latex] because it is the largest number that divides evenly into both [latex]16[/latex] and [latex]20[/latex] The GCF of polynomials works the same way: [latex]4x[/latex] is the GCF of [latex]16x[/latex] and [latex]20{x}^{2}[/latex] because it is the largest polynomial that divides evenly into both [latex]16x[/latex] and [latex]20{x}^{2}[/latex].

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

A General Note: Greatest Common Factor

The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.

How To: Given a polynomial expression, factor out the greatest common factor.

  1. Identify the GCF of the coefficients.
  2. Identify the GCF of the variables.
  3. Combine to find the GCF of the expression.
  4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.
  5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.

Example: Factoring the Greatest Common Factor

Factor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[/latex].

Try It

Factor [latex]x\left({b}^{2}-a\right)+6\left({b}^{2}-a\right)[/latex] by pulling out the GCF.

Watch this video to see more examples of how to factor the GCF from a trinomial.

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

Trinomials of the form [latex]{x}^{2}+bx+c[/latex] can be factored by finding two numbers with a product of [latex]c[/latex] and a sum of [latex]b[/latex]. The trinomial [latex]{x}^{2}+10x+16[/latex], for example, can be factored using the numbers [latex]2[/latex] and [latex]8[/latex] because the product of those numbers is [latex]16[/latex] and their sum is [latex]10[/latex]. The trinomial can be rewritten as the product of [latex]\left(x+2\right)[/latex] and [latex]\left(x+8\right)[/latex].

A General Note: Factoring a Trinomial with Leading Coefficient 1

A trinomial of the form [latex]{x}^{2}+bx+c[/latex] can be written in factored form as [latex]\left(x+p\right)\left(x+q\right)[/latex] where [latex]pq=c[/latex] and [latex]p+q=b[/latex].

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 [latex]{x}^{2}+bx+c[/latex], factor it.

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

Example: Factoring a Trinomial with Leading Coefficient 1

Factor [latex]{x}^{2}+2x - 15[/latex].

Q & A

Does the order of the factors matter?

No. Multiplication is commutative, so the order of the factors does not matter.

Try It

Factor [latex]{x}^{2}-7x+6[/latex].

Factoring by Grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial [latex]2{x}^{2}+5x+3[/latex] can be rewritten as [latex]\left(2x+3\right)\left(x+1\right)[/latex] using this process. We begin by rewriting the original expression as [latex]2{x}^{2}+2x+3x+3[/latex] and then factor each portion of the expression to obtain [latex]2x\left(x+1\right)+3\left(x+1\right)[/latex]. We then pull out the GCF of [latex]\left(x+1\right)[/latex] to find the factored expression.

A General Note: Factor by Grouping

To factor a trinomial in the form [latex]a{x}^{2}+bx+c[/latex] by grouping, we find two numbers with a product of [latex]ac[/latex] and a sum of [latex]b[/latex]. We use these numbers to divide the [latex]x[/latex] term into the sum of two terms and factor each portion of the expression separately, then factor out the GCF of the entire expression.

How To: Given a trinomial in the form [latex]a{x}^{2}+bx+c[/latex], factor by grouping.

  1. List factors of [latex]ac[/latex].
  2. Find [latex]p[/latex] and [latex]q[/latex], a pair of factors of [latex]ac[/latex] with a sum of [latex]b[/latex].
  3. Rewrite the original expression as [latex]a{x}^{2}+px+qx+c[/latex].
  4. Pull out the GCF of [latex]a{x}^{2}+px[/latex].
  5. Pull out the GCF of [latex]qx+c[/latex].
  6. Factor out the GCF of the expression.

Example: Factoring a Trinomial by Grouping

Factor [latex]5{x}^{2}+7x - 6[/latex] by grouping.

Try It

Factor the following.

  1. [latex]2{x}^{2}+9x+9[/latex]
  2. [latex]6{x}^{2}+x - 1[/latex]

In the next video we show another example of how to factor a trinomial by grouping.