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.[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]
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].
Example 4: Factoring a Perfect Square Trinomial
Factor [latex]25{x}^{2}+20x+4[/latex].Solution
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]. Then check to see if the middle term is twice the product of [latex]5x[/latex] and [latex]2[/latex]. The middle term is, indeed, twice the product: [latex]2\left(5x\right)\left(2\right)=20x[/latex]. Therefore, the trinomial is a perfect square trinomial and can be written as [latex]{\left(5x+2\right)}^{2}[/latex].Try It 4
Factor [latex]49{x}^{2}-14x+1[/latex]. SolutionFactoring 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]
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].
Example 5: Factoring a Difference of Squares
Factor [latex]9{x}^{2}-25[/latex].Solution
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]. The polynomial represents a difference of squares and can be rewritten as [latex]\left(3x+5\right)\left(3x - 5\right)[/latex].Try It 5
Factor [latex]81{y}^{2}-100[/latex]. SolutionQ & A
Is there a formula to factor the sum of squares?
No. A sum of squares cannot be factored.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]
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]
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.
- Confirm that the first and last term are cubes, [latex]{a}^{3}+{b}^{3}[/latex] or [latex]{a}^{3}-{b}^{3}[/latex].
- 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].