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]
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- Screenshot: Method to the Madness. Provided by: Lumen Learning License: CC BY: Attribution.
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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.