Skills Review for Arc Length of a Curve and Surface Area

Learning Outcomes

  • Factor a perfect square trinomial

In the Arc Length of a Curve and Surface Area section, some of the integrals that we set up can be solved using substitution or taking the square root of a factored perfect square trinomial. Here we will review how to factor a perfect square trinomial.

Factor 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]
[latex]\\[/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

  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].

Example: Factoring a Perfect Square Trinomial

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

Try It

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

try it