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 the figure below.

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 studied fractions, we learned 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 is to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

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 these 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].

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.

Factoring Special Cases

Factoring a Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

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]\\[/latex]

We can use this equation to factor any differences of squares.

Factoring the Sum and Difference of Cubes

Now we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

[latex]\\[/latex]

Similarly, the sum of cubes can be factored into a binomial and a trinomial but with different signs.

[latex]\\[/latex]

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: Same Opposite Always Positive. For example, consider the following example.

The sign of the first 2 is the same as the sign between [latex]{x}^{3}-{2}^{3}[/latex]. The sign of the [latex]2x[/latex] term is opposite the sign between [latex]{x}^{3}-{2}^{3}[/latex]. And the sign of the last term, 4, is always positive.

Factoring 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].

Key Equations

Key Concepts

• 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 of 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