# 13.2: Breaking Up is NOT Hard to Do

**At Grade**Created by: CK-12

*This activity is intended to supplement Algebra I, Chapter 12, Lesson 6.*

## Problem 1 – Introduction

Press \begin{align*}Y=\end{align*}

\begin{align*}Y1(x) = \frac{7x+3}{x^2 - 9}\end{align*}

Also, for the second expression, move the cursor to the left of \begin{align*}Y2 =\end{align*}**ENTER** until a circle appears. This will place a large circle in front of the graph as it is graphed on the handheld.

To view the graphs, press **ZOOM** and select **ZStandard**.

1. How do the graphs of the two given equations compare?

2. What do the graphic results tell us about the two functions?

Functions can often be expressed in several different ways. The second representation splits the initial rational function into fractional parts and is referred to as the **sum of partial fractions**.

3. How are the denominators in \begin{align*}\frac{3}{x+3} + \frac{4}{x-3}\end{align*}

So, to begin understanding how these partial fractions are developed, begin by writing two fractions using the factors of the denominator of \begin{align*}Y1\end{align*}

\begin{align*}Y1(x) & = Y2(x) \\
\frac{7x+3}{x^2-9} & = \frac{A}{x+3} + \frac{B}{x-3}\end{align*}

4. What is the LCD (least common denominator) for \begin{align*}\frac{7x+3}{x^2-9} = \frac{A}{x+3} + \frac{B}{x-3}\end{align*}

5. What is the result of multiplying both sides of \begin{align*}\frac{7x+3}{x^2-9} = \frac{A}{x+3} + \frac{B}{x-3}\end{align*}

6. Substitute in a convenient number for \begin{align*}x\end{align*}

7. Similarly substitute in a convenient number for \begin{align*}x\end{align*}

8. Now substitute the values you found for both \begin{align*}A\end{align*}

9. How do your results for Question 8 support your answer to the Question 2 regarding what the graphs of the functions \begin{align*}Y1\end{align*}

## Problem 2 – Practice

10. Express the rational function, \begin{align*}f(x) = \frac{7x-4}{x^2 + x-6}\end{align*}, as a sum of partial fractions.

11. Graph the initial function and your sum of partial fractions using the graphing calculator as outlined in Problem 1. How does this verify your results? Explain your reasoning.

## Problem 3 – The Next Level

12. Express the rational function, \begin{align*}f(x) = \frac{5x-7}{4x^2 - 8x-12}\end{align*}, as a sum of partial fractions.

13. Graph the initial function and your sum of partial fractions using the graphing calculator. How does this verify your results? Explain you reasoning.

**Additional Practice Problems**

Represent each of the following rational functions as a sum of partial fractions. Verify your results graphically.

14. \begin{align*}f(x) = \frac{-7x-11}{x^2 + 4x+3}\end{align*}

15. \begin{align*}f(x) = \frac{2x+42}{x^2+2x-24}\end{align*}

16. \begin{align*}f(x) = \frac{x}{x^2+2x-8}\end{align*}