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?

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

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.

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

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

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 3 +3 3𝑛 2 ∗ 5 5 24𝑛 15 𝑛 2 − 5 𝑛 3 +15 15𝑛 2 =− 5𝑛 3 −24𝑛+15 15 𝑛 2

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+5 + 9 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)

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)

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