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Study Guides > Intermediate Algebra

Read: Special Cases - Squares

Learning Objectives

  • Factor a polynomial of the form: [latex]{a}^{2}+2ab+{b}^{2}[/latex]
  • Factor a polynomial of the form: [latex]{a}^{2}-{b}^{2}[/latex]
Picture of a sidewalk leading to a parking lot. There is a path through the grass to the right of the sidewalk through the trees that has been made by people walking on the grass. The shortcut to the parking lot is the preferred way. Some people find it helpful to know when they can take a shortcut to avoid doing extra work.  There are some polynomials that will always factor a certain way, and for those we offer a shortcut.  Most people find it helpful to memorize the factored form of a perfect square trinomial or a difference of squares.  The most important skill you will use in this section will be recognizing when you can use the shortcuts.

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]
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]
In the following example we will show you how to define a, and b so you can use the shortcut.

Exercises

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

Answer: First, notice that [latex]25{x}^{2}[/latex] and [latex]4[/latex] are perfect squares because [latex]25{x}^{2}={\left(5x\right)}^{2}[/latex] and [latex]4={2}^{2}[/latex]. This means that [latex]a=5x\text{ and }b=2[/latex] Next, check to see if the middle term is equal to [latex]2ab[/latex], which it is:

[latex]2ab = 2\left(5x\right)\left(2\right)=20x[/latex].

  Therefore, the trinomial is a perfect square trinomial and can be written as [latex]{\left(a+b\right)}^{2}={\left(5x+2\right)}^{2}[/latex].

Answer

[latex-display]25{x}^{2}+20x+4={\left(5x+2\right)}^{2}[/latex-display]

In the next example, we will show that we can use [latex]1 = 1^2[/latex] to factor a polynomial with a term equal to [latex]1[/latex].

Example

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

Answer: First, notice that [latex]49{x}^{2}[/latex] and [latex]1[/latex] are perfect squares because [latex]49{x}^{2}={\left(7x\right)}^{2}[/latex] and [latex]1={1}^{2}[/latex]. This means that [latex]a=7x[/latex], we could say that [latex]b=1[/latex], but would that give a middle term of [latex]-14x[/latex]? We will need to choose [latex]b = -1[/latex] to get the results we want:

[latex]2ab = 2\left(7x\right)\left(-1\right)=-14x[/latex].

Therefore, the trinomial is a perfect square trinomial and can be written as [latex]{\left(a-b\right)}^{2}={\left(7x-1\right)}^{2}[/latex].

Answer

[latex-display]49{x}^{2}-14x+1={\left(7x-1\right)}^{2}[/latex-display]  

In the following video we provide another short description of what a perfect square trinomial is, and show how to factor them using a the formula. https://youtu.be/UMCVGDTxxTI We can summarize our process in the following way:

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], or[latex]{\left(a-b\right)}^{2}[/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]
Tip:  This is a very common pattern to factor, so become very familiar with this one.

Example

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]. This means that [latex]a=3x,\text{ and }b=5[/latex] The polynomial represents a difference of squares and can be rewritten as [latex]\left(3x+5\right)\left(3x - 5\right)[/latex]. Check that you are correct by multiplying. [latex-display]\left(3x+5\right)\left(3x - 5\right)=9x^2-15x+15x-25=9x^2-25[/latex-display]

Answer

[latex-display]9{x}^{2}-25=\left(3x+5\right)\left(3x - 5\right)[/latex-display]

The most helpful thing for recognizing a difference of squares that can be factored with the shortcut is knowing which numbers are perfect squares, as you will see in the next example.

Example

Factor [latex]81{y}^{2}-144[/latex].

Answer: Notice that [latex]81{y}^{2}[/latex] and [latex]144[/latex] are perfect squares because [latex]81{y}^{2}={\left(9x\right)}^{2}[/latex] and [latex]144={12}^{2}[/latex]. This means that [latex]a=9x,\text{ and }b=12[/latex] The polynomial represents a difference of squares and can be rewritten as [latex]\left(9x+12\right)\left(9x - 12\right)[/latex]. Check that you are correct by multiplying. [latex-display]\left(9x+12\right)\left(9x - 12\right)=81x^2-108x+108x-144=81x^2-144[/latex-display]

Answer

[latex-display]81{y}^{2}-144=\left(9x+12\right)\left(9x - 12\right)[/latex-display]

 
In the following video we show another example of how to use the formula for fact a difference of squares. https://youtu.be/Li9IBp5HrFA We can summarize the process for factoring a difference of squares with the shortcut this way:

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

  1. Confirm that the first and last term are both perfect squares. For exponents, this means the exponent must be divisible by 2.
  2. Write the factored form as [latex]\left(a+b\right)\left(a-b\right)[/latex].

Think About It

Is there a formula to factor the sum of squares, [latex]a^2+b^2[/latex], into a product of two binomials? Write down some ideas for how you would answer this in the box below before you look at the answer. [practice-area rows="1"][/practice-area]

Answer: There is no way to factor a sum of squares into a product of two binomials, this is because of addition - the middle term needs to "disappear" and the only way to do that is with opposite signs.  to get a positive result, you must multiply two numbers with the same signs. The only time a sum of squares can be factored is if they share any common factors, as in the following case: [latex-display]9x^2+36[/latex-display] [latex-display]9x^2={(3x)}^2, \text{ and }36 = 6^2[/latex-display] The only way to factor this expression is by pulling out the GCF which is 9. [latex-display]9x^2+36=9(x^2+4)[/latex-display]

   

Licenses & Attributions

CC licensed content, Original

CC licensed content, Shared previously

  • Factor Perfect Square Trinomials Using a Formula. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Ex: Factor a Difference of Squares. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Unit 12: Factoring, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.