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.

\begin{array}{ccc} a^{2} + 2 a b + b^{2} & = & {(a + b)}^{2} \\ and \\ a^{2} - 2 a b + b^{2} & = & {(a - b)}^{2} \end{array}

$\begin{array}{ccc}\hfill {a}^{2}+2ab+{b}^{2}& =& {\left(a+b\right)}^{2}\hfill \\ & \text{and}& \\ \hfill {a}^{2}-2ab+{b}^{2}& =& {\left(a-b\right)}^{2}\hfill \end{array}$

$\\$

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:

a^{2} + 2 a b + b^{2} = {(a + b)}^{2}

${a}^{2}+2ab+{b}^{2}={\left(a+b\right)}^{2}$

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 $ab$ .
Write the factored form as ${\left(a+b\right)}^{2}$ .

Example: Factoring a Perfect Square Trinomial

Factor $25{x}^{2}+20x+4$ .

Show Solution

Notice that $25{x}^{2}$ and $4$ are perfect squares because $25{x}^{2}={\left(5x\right)}^{2}$ and $4={2}^{2}$ . Then check to see if the middle term is twice the product of $5x$ and $2$ . The middle term is, indeed, twice the product: $2\left(5x\right)\left(2\right)=20x$ . Therefore, the trinomial is a perfect square trinomial and can be written as ${\left(5x+2\right)}^{2}$ .

Try It

Factor $49{x}^{2}-14x+1$ .

Show Solution

${\left(7x - 1\right)}^{2}$

Applications of Integration: Prerequisite Material