1. Section 8.2 – Adding and Subtracting Rational Expressions
EQ: How do I use my prior knowledge about adding and subtracting fractions and apply it to adding and subtracting rational expressions?

2. How do you add fractions…?
Find a common denominator
Add the numerators

3. LCM – Least Common Multiple
To add or subtract two rational expressions that have an unlike denominators, you must first find the least common denominator (LCD) To find the LCM of two or more numbers, factor them. The LCM contains each factor the greatest number of times it appears as a factor.

4. Example 1 Find the LCM of x3 – x2 – 2x and x2 – 4x + 4.
Step 1: Factor each polynomial
x3 – x2 – 2x x2 – 4x + 4
𝑥(𝑥−2)(𝑥+1) 𝑥−2 𝑥−2 = (𝑥−2) 2
Step 2: Find the LCM
𝑥(𝑥+1)( 𝑥−2) 2

5. Example 2 Find the LCM of x3 + 2x2 – 3x and x2 + 6x + 9.
Step 1: Factor each polynomial
x3 + 2x2 – 3x x2 + 6x + 9
𝑥(𝑥+3)(𝑥−1) 𝑥+3 𝑥+3 = (𝑥+3) 2
Step 2: Find the LCM
𝑥(𝑥−1)( 𝑥+3) 2

6. Example 3 Simplify 8 5𝑛 − 𝑛 3 +3 3𝑛 2 . LCM:
Multiply the numerator and the denominator by what it is missing from the LCM.
15 𝑛 2
8 5𝑛 ∗ 3𝑛 3𝑛 − 𝑛 𝑛 2 ∗ 5 5
24𝑛 15 𝑛 2 − 5 𝑛 𝑛 2 =− 5𝑛 3 −24𝑛 𝑛 2

7. Example 4 Simplify 8 𝑥 2 −6𝑥−16 + 9 𝑥 2 −3𝑥−40 LCM:
= 8 (𝑥−8)(𝑥+2) + 9 (𝑥−8)(𝑥+5)
(𝑥−8)(𝑥+2)(𝑥+5)
= 8 (𝑥−8)(𝑥+2) ∗ 𝑥+5 𝑥 𝑥−8 𝑥+5 ∗ 𝑥+2 𝑥+2
= 8𝑥+40 (𝑥−8)(𝑥+2)(𝑥+5) + 9𝑥+18 (𝑥−8)(𝑥+2)(𝑥+5)
= 17𝑥+58 (𝑥−8)(𝑥+2)(𝑥+5)

8. Example 5 Simplify 𝑥+10 𝑥 2 −9 − 𝑥 𝑥−3 LCM: 𝑥+10 (𝑥+3)(𝑥−3) − 𝑥 𝑥−3
(𝑥−3)(𝑥+3)
𝑥+10 (𝑥+3)(𝑥−3) − 𝑥 𝑥−3 ∗ 𝑥+3 𝑥+3 = 𝑥+10 𝑥−3 𝑥+3 − 𝑥 2 +3𝑥 (𝑥−3)(𝑥+3)
𝑥+10−( 𝑥 2 +3𝑥) (𝑥−3)(𝑥+3) = − 𝑥 2 −2𝑥+10 (𝑥−3)(𝑥+3)

9. Example 6 Simplify 𝑥+5 2𝑥−4 − 3𝑥+8 4𝑥−8 = 𝑥+5 2(𝑥−2) − 3𝑥+8 4(𝑥−2)
= 𝑥+5 2(𝑥−2) ∗ 2 2 − 3𝑥+8 4(𝑥−2)
= 2(𝑥+5) 4(𝑥−2) − 3𝑥+8 4 𝑥−2 = 2𝑥+10− 3𝑥+8 4 𝑥−2 = −𝑥+2 4 𝑥−2 = − 𝑥−2 4 𝑥−2 =− 1 4