▪ Factoring Special Cases

Learning Outcomes

  • Factor a perfect square trinomial.
  • Factor a difference of squares.
  • Factor a sum and difference of cubes.
  • Factor an expression with negative or fractional exponents.

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.

[latex]\begin{array}{ccc}\hfill {a}^{2}+2ab+{b}^{2}& =& {\left(a+b\right)}^{2}\hfill \\ & \text{and}& \\ \hfill {a}^{2}-2ab+{b}^{2}& =& {\left(a-b\right)}^{2}\hfill \end{array}[/latex]
[latex]\\[/latex]
We can use this equation to factor any perfect square trinomial.

A General Note: Perfect Square Trinomials

A perfect square trinomial can be written as the square of a binomial:

[latex]{a}^{2}+2ab+{b}^{2}={\left(a+b\right)}^{2}[/latex]

How To: Given a perfect square trinomial, factor it into the square of a binomial

  1. Confirm that the first and last term are perfect squares.
  2. Confirm that the middle term is twice the product of [latex]ab[/latex].
  3. Write the factored form as [latex]{\left(a+b\right)}^{2}[/latex].

Example: Factoring a Perfect Square Trinomial

Factor [latex]25{x}^{2}+20x+4[/latex].

Try It

Factor [latex]49{x}^{2}-14x+1[/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]

[latex]\\[/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].

Try It

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

Q & A

Is there a formula to factor the sum of squares?

No. A sum of squares cannot be factored.

Watch this video to see another example of how to factor a difference 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]{a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)[/latex]

[latex]\\[/latex]

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

[latex]{a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)[/latex]

[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.

[latex]{x}^{3}-{2}^{3}=\left(x - 2\right)\left({x}^{2}+2x+4\right)[/latex]

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.

A General Note: Sum and Difference of Cubes

We can factor the sum of two cubes as

[latex]{a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)[/latex]

We can factor the difference of two cubes as

[latex]{a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)[/latex]

How To: Given a sum of cubes or difference of cubes, factor it

  1. Confirm that the first and last term are cubes, [latex]{a}^{3}+{b}^{3}[/latex] or [latex]{a}^{3}-{b}^{3}[/latex].
  2. For a sum of cubes, write the factored form as [latex]\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)[/latex]. For a difference of cubes, write the factored form as [latex]\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)[/latex].

Example: Factoring a Sum of Cubes

Factor [latex]{x}^{3}+512[/latex].

Try It

Factor the sum of cubes [latex]216{a}^{3}+{b}^{3}[/latex].

Example: Factoring a Difference of Cubes

Factor [latex]8{x}^{3}-125[/latex].

Try It

Factor the difference of cubes: [latex]1,000{x}^{3}-1[/latex].

In the following two video examples we show more binomials that can be factored as a sum or difference of cubes.

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].

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].

The following video provides more examples of how to factor expressions with negative exponents.

Try It

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


The following video provides more examples of how to factor expressions with fractional exponents.