Power and Polynomial Functions: Background You’ll Need 1

  • Solve equations by factoring differences of squares and perfect square trinomials

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.

perfect square trinomial

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].
Factor [latex]25{x}^{2}+20x+4[/latex].SolutionTo factor the quadratic expression [latex]25{x}^{2}+20x+4[/latex], recognizing it as a perfect square trinomial will streamline the process. This type of expression comes from squaring a binomial and has a special format, [latex]a^2 +2ab+b^2[/latex], where it can be rewritten as [latex](a+b)^2[/latex].

[latex]\begin{align*} \text{Original expression:} & \quad 25x^2 + 20x + 4 \\ \text{Identify square terms:} & \quad 25x^2 = (5x)^2 \quad \text{and} \quad 4 = 2^2 \\ \text{Check middle term:} & \quad 2 \times 5x \times 2 = 20x \\ \text{Write as a square of a binomial:} & \quad (5x + 2)^2 \end{align*}[/latex]
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]

We can use this equation to factor any differences of squares.

difference 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].
Factor [latex]9{x}^{2}-25[/latex].SolutionTo factor the quadratic expression [latex]9{x}^{2}-25[/latex], we recognize that it is a difference of squares.

[latex]\begin{align*} \text{Original expression:} & \quad 9x^2 - 25 \\ \text{Identify square terms:} & \quad 9x^2 = (3x)^2 \quad \text{and} \quad 25 = 5^2 \\ \text{Apply difference of squares formula:} & \quad (3x + 5)(3x - 5) \end{align*}[/latex]
Factor [latex]81{y}^{2}-100[/latex].