# Special Cases - Squares

### Learning Outcomes

- Factor special products

## 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.### A General Note: Perfect Square Trinomials

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

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].### 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] and [latex]b=1[/latex]. Next, check to see if the middle term is equal to [latex]2ab[/latex], which it is:

[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].### Try It

[ohm_question]91970[/ohm_question]### How To: Given a perfect square trinomial, factor it into the square of a binomial**
**

- Confirm that the first and last term are perfect squares.
- Confirm that the middle term is twice the product of [latex]ab[/latex].
- 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. This type of polynomial is unique because it can be factored into two binomials but has only two terms.### Factor a Difference of Squares

Given [latex]a^2-b^2[/latex], its factored form will be [latex]\left(a+b\right)\left(a-b\right)[/latex].Multiply:

[latex]\begin{array}{l}\left(x-2\right)\left(x+2\right)\\\text{}\\=x^2-2x+2x-2^2\\\text{}\\=x^2-2^2\\\text{}\\=x^2-4\end{array}[/latex]

The polynomial [latex]x^2-4[/latex] is called a difference of squares because each term can be written as something squared. A difference of squares will always factor in the following way: Let’s factor [latex]x^{2}–4[/latex] by writing it as a trinomial, [latex]x^{2}+0x–4[/latex]. This is similar in format to the trinomials we have been factoring so far, so let’s use the same method.Find the factors of [latex]a\cdot{c}[/latex] whose sum is *b, *in this case, 0*:*

Factors of [latex]−4[/latex] | Sum of the factors |
---|---|

[latex]1\cdot-4=−4[/latex] | [latex]1-4=−3[/latex] |

[latex]2\cdot−2=−4[/latex] | [latex]2-2=0[/latex] |

[latex]-1\cdot4=−4[/latex] | [latex]-1+4=3[/latex] |

### Example

Factor [latex]x^{2}–4[/latex].Answer: Rewrite [latex]0x[/latex] as [latex]−2x+2x[/latex].

[latex]\begin{array}{l}x^{2}+0x-4\\x^{2}-2x+2x-4\end{array}[/latex]

Group pairs.[latex]\left(x^{2}–2x\right)+\left(2x–4\right)[/latex]

Factor*x*out of the first group. Factor [latex]2[/latex] out of the second group.

[latex]x\left(x–2\right)+2\left(x–2\right)[/latex]

Factor out [latex]\left(x–2\right)[/latex].[latex]\left(x–2\right)\left(x+2\right)[/latex]

#### Answer

[latex-display]\left(x–2\right)\left(x+2\right)[/latex-display]### A General Note: Differences of Squares

A difference of squares can be rewritten as two factors containing the same terms but opposite signs.### 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]

### Try It

[ohm_question]161674[/ohm_question]### 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]

1^2 | 1 |

2^2 | 4 |

3^2 | 9 |

4^2 | 16 |

5^2 | 25 |

### Try It

[ohm_question]7930[/ohm_question]### How To: Given a difference of squares, factor it into binomials

- Confirm that the first and last term are perfect squares.
- 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] 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]

## Contribute!

## Licenses & Attributions

### CC licensed content, Original

- Screenshot: Method to the Madness.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Image: Shortcut This Way.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution.

### 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.