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Study Guides > College Algebra

Factoring Polynomials

Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in Figure 1.
A large rectangle with smaller squares and a rectangle inside. The length of the outer rectangle is 6x and the width is 10x. The side length of the squares is 4 and the height of the width of the inner rectangle is 4. Figure 1
The area of the entire region can be found using the formula for the area of a rectangle.

[latex]\begin{array}{ccc}\hfill A& =& lw\hfill \\ & =& 10x\cdot 6x\hfill \\ & =& 60{x}^{2}{\text{ units}}^{2}\hfill \end{array}[/latex]

The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area of [latex]A={s}^{2}={4}^{2}=16[/latex] units2. The other rectangular region has one side of length [latex]10x - 8[/latex] and one side of length [latex]4[/latex], giving an area of [latex]A=lw=4\left(10x - 8\right)=40x - 32[/latex] units2. So the region that must be subtracted has an area of [latex]2\left(16\right)+40x - 32=40x[/latex] units2. The area of the region that requires grass seed is found by subtracting [latex]60{x}^{2}-40x[/latex] units2. This area can also be expressed in factored form as [latex]20x\left(3x - 2\right)[/latex] units2. We can confirm that this is an equivalent expression by multiplying. Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of methods that can be used to factor polynomial expressions.

Factoring Basics

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].

Answer: First, find the GCF of the expression. The GCF of [latex]6,45[/latex], and [latex]21[/latex] is [latex]3[/latex]. The GCF of [latex]{x}^{3},{x}^{2}[/latex], and [latex]x[/latex] is [latex]x[/latex]. (Note that the GCF of a set of expressions in the form [latex]{x}^{n}[/latex] will always be the exponent of lowest degree.) And the GCF of [latex]{y}^{3},{y}^{2}[/latex], and [latex]y[/latex] is [latex]y[/latex]. Combine these to find the GCF of the polynomial, [latex]3xy[/latex]. Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that [latex]3xy\left(2{x}^{2}{y}^{2}\right)=6{x}^{3}{y}^{3},3xy\left(15xy\right)=45{x}^{2}{y}^{2}[/latex], and [latex]3xy\left(7\right)=21xy[/latex]. Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.

[latex]\left(3xy\right)\left(2{x}^{2}{y}^{2}+15xy+7\right)[/latex]

Analysis of the Solution

After factoring, we can check our work by multiplying. Use the distributive property to confirm that [latex]\left(3xy\right)\left(2{x}^{2}{y}^{2}+15xy+7\right)=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.

Answer: [latex]\left({b}^{2}-a\right)\left(x+6\right)[/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.

Answer: We have a trinomial with [latex]a=5,b=7[/latex], and [latex]c=-6[/latex]. First, determine [latex]ac=-30[/latex]. We need to find two numbers with a product of [latex]-30[/latex] and a sum of [latex]7[/latex]. In the table, we list factors until we find a pair with the desired sum.

Factors of [latex]-30[/latex] Sum of Factors
[latex]1,-30[/latex] [latex]-29[/latex]
[latex]-1,30[/latex] 29
[latex]2,-15[/latex] [latex]-13[/latex]
[latex]-2,15[/latex] 13
[latex]3,-10[/latex] [latex]-7[/latex]
[latex]-3,10[/latex] 7
So [latex]p=-3[/latex] and [latex]q=10[/latex].
[latex]\begin{array}{cc}5{x}^{2}-3x+10x - 6 \hfill & \text{Rewrite the original expression as }a{x}^{2}+px+qx+c.\hfill \\ x\left(5x - 3\right)+2\left(5x - 3\right)\hfill & \text{Factor out the GCF of each part}.\hfill \\ \left(5x - 3\right)\left(x+2\right)\hfill & \text{Factor out the GCF}\text{ }\text{ of the expression}.\hfill \end{array}[/latex]

Analysis of the Solution

We can check our work by multiplying. Use FOIL to confirm that [latex]\left(5x - 3\right)\left(x+2\right)=5{x}^{2}+7x - 6[/latex].

Try It

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

Answer:

  1. [latex]\left(2x+3\right)\left(x+3\right)[/latex]
  2. [latex]\left(3x - 1\right)\left(2x+1\right)[/latex]

Factoring a Difference of Squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.
[latex]{a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)[/latex]
We can use this equation to factor any differences of squares.

A General Note: Differences of Squares

A difference of squares can be rewritten as two factors containing the same terms but opposite signs.
[latex]{a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)[/latex]

How To: Given a difference of squares, factor it into binomials.

  1. Confirm that the first and last term are perfect squares.
  2. Write the factored form as [latex]\left(a+b\right)\left(a-b\right)[/latex].

Example: Factoring a Difference of Squares

Factor [latex]9{x}^{2}-25[/latex].

Answer: Notice that [latex]9{x}^{2}[/latex] and [latex]25[/latex] are perfect squares because [latex]9{x}^{2}={\left(3x\right)}^{2}[/latex] and [latex]25={5}^{2}[/latex]. The polynomial represents a difference of squares and can be rewritten as [latex]\left(3x+5\right)\left(3x - 5\right)[/latex].

Try It

Factor [latex]81{y}^{2}-100[/latex].

Answer: [latex-display]\left(9y+10\right)\left(9y - 10\right)[/latex-display]

Factor Expressions with Fractional or Negative Exponents

Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance, [latex]2{x}^{\frac{1}{4}}+5{x}^{\frac{3}{4}}[/latex] can be factored by pulling out [latex]{x}^{\frac{1}{4}}[/latex] and being rewritten as [latex]{x}^{\frac{1}{4}}\left(2+5{x}^{\frac{1}{2}}\right)[/latex].

Example: Factoring an Expression with Fractional or Negative Exponents

Factor [latex]3x{\left(x+2\right)}^{\frac{-1}{3}}+4{\left(x+2\right)}^{\frac{2}{3}}[/latex].

Answer: Factor out the term with the lowest value of the exponent. In this case, that would be [latex]{\left(x+2\right)}^{-\frac{1}{3}}[/latex].

[latex]\begin{array}{cc}{\left(x+2\right)}^{-\frac{1}{3}}\left(3x+4\left(x+2\right)\right)\hfill & \text{Factor out the GCF}.\hfill \\ {\left(x+2\right)}^{-\frac{1}{3}}\left(3x+4x+8\right)\hfill & \text{Simplify}.\hfill \\ {\left(x+2\right)}^{-\frac{1}{3}}\left(7x+8\right)\hfill & \end{array}[/latex]

Try It

Factor [latex]2{\left(5a - 1\right)}^{\frac{3}{4}}+7a{\left(5a - 1\right)}^{-\frac{1}{4}}[/latex].

Answer: [latex]{\left(5a - 1\right)}^{-\frac{1}{4}}\left(17a - 2\right)[/latex]

Key Equations

 
difference of squares [latex]{a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)[/latex]
perfect square trinomial [latex]{a}^{2}+2ab+{b}^{2}={\left(a+b\right)}^{2}[/latex]
sum of cubes [latex]{a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)[/latex]
difference of cubes [latex]{a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)[/latex]
  • The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem.
  • Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term.
  • Trinomials can be factored using a process called factoring by grouping.
  • Perfect square trinomials and the difference of squares are special products and can be factored using equations.
  • The sum of cubes and the difference of cubes can be factored using equations.
  • Polynomials containing fractional and negative exponents can be factored by pulling out a GCF.

Glossary

factor by grouping a method for factoring a trinomial in the form [latex]a{x}^{2}+bx+c[/latex] 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 greatest common factor the largest polynomial that divides evenly into each polynomial

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  • College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
  • Example: Greatest Common Factor. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Factoring Trinomials by Grouping. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Question ID 7886, 7897, 7908, 7919, 7929, 7922. Authored by: Tyler Wallace. License: CC BY: Attribution. License terms: IMathAS Community License CC- BY + GPL.
  • Examples: Factoring Binomials (Special). Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Ex1: Factor a Sum or Difference of Cubes. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Ex3: Factor a Sum or Difference of Cubes. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Question ID 93666, 93668. Authored by: Michael Jenck. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.

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