Adding and Subtracting Rational Expressions Section 7.4 beginning on page 384

Adding/Subtracting With Like Denominators When the denominators are the same, add or subtract the numerator and keep the denominator. Example 1: a) 7 4𝑥 + 3 4𝑥 b) 2𝑥 𝑥+6 − 5 𝑥+6 = 7+3 4𝑥 = 10 4𝑥 = 5 2𝑥 = 2𝑥−5 𝑥+6

Finding the Least Common Multiple Example 2: Find the least common multiple of 4 𝑥 2 −16 and 6 𝑥 2 −24𝑥+24 ** Factor each polynomial and write numerical factors as products of primes. 4 𝑥 2 −16 6 𝑥 2 −24𝑥+24 =4( 𝑥 2 −4) =6( 𝑥 2 −4𝑥+4) =2∙3(𝑥−2)(𝑥−2) = 2 2 (𝑥+2)(𝑥−2) =2∙3 (𝑥−2) 2 **The LCM is the product of the highest power of each factor that appears in either polynomial. 𝐿𝐶𝑀= 2 2 ∙3(𝑥+2) (𝑥−2) 2 =12(𝑥+2) (𝑥−2) 2

Adding with Unlike Denominators Example 3: Find the sum 7 9 𝑥 2 + 𝑥 3 𝑥 2 +3𝑥 Step 1: Create equivalent fractions that have the same denominator. You can always find a common denominator by multiplying the denominators together. If you use the LCM simplifying at the end may be easier but the creating equivalent fractions will be harder. 3 𝑥 2 +3𝑥 3 𝑥 2 +3𝑥 ∙ 7 9 𝑥 2 + 𝑥 3 𝑥 2 +3𝑥 ∙ 9 𝑥 2 9 𝑥 2 = 7(3 𝑥 2 +3𝑥) 9 𝑥 2 (3 𝑥 2 +3𝑥) + 9 𝑥 2 (𝑥) 9𝑥 2 (3 𝑥 2 +3𝑥) Step 2: Add the numerators, simplify the numerator. Step 3: Factor the numerator and denominator. Step 4: Simplify. = 7 3 𝑥 2 +3𝑥 +9 𝑥 2 (𝑥) 9 𝑥 2 (3 𝑥 2 +3𝑥) = 9 𝑥 3 +21 𝑥 2 +21𝑥 9 𝑥 2 (3 𝑥 2 +3𝑥) = 3𝑥(3 𝑥 2 +7𝑥+7) 9 𝑥 2 ∙3𝑥(𝑥+1) = 3 𝑥 2 +7𝑥+7 9 𝑥 2 (𝑥+1)

Subtracting With Unlike Denominators Example 4: Find the difference 𝑥+2 2𝑥−2 − −2𝑥−1 𝑥 2 −4𝑥−3 It can be helpful to factor the denominators first to allow you to multiply the fractions by fewer factors. (by only multiplying by the factors that are not already in both denominators) = 𝑥+2 2(𝑥−1) − −2𝑥−1 (𝑥−3)(𝑥−1) (𝑥−3) (𝑥−3) ∙ ∙ 2 2 = 𝑥 2 −𝑥−6 2(𝑥−3)(𝑥−1) − −4𝑥−2 2(𝑥−3)(𝑥−1) = 𝑥 2 −𝑥−6−(−4𝑥−2) 2(𝑥−3)(𝑥−1) = 𝑥 2 −𝑥−6+4𝑥+2 2(𝑥−3)(𝑥−1) = 𝑥 2 +3𝑥−4 2(𝑥−3)(𝑥−1) = (𝑥−1)(𝑥+4) 2(𝑥−3)(𝑥−1) = 𝑥+4 2(𝑥−3) ;𝑥≠1

Simplifying Complex Fractions 5 𝑥+4 1 𝑥+4 + 2 𝑥 Example 6: Simplify 5 𝑥+4 1 𝑥+4 + 2 𝑥 Step 1: Simplify the numerator and denominator so that there is only one fraction in each. Step 2: To divide, multiply the numerator by the reciprocal of the denominator. (𝑥+4) (𝑥+4) 𝑥 𝑥 ∙ = 5 𝑥+4 𝑥 𝑥(𝑥+4) + 2𝑥+8 𝑥(𝑥+4) = 5 𝑥+4 3𝑥+8 𝑥(𝑥+4) = 5 𝑥+4 ∙ 𝑥(𝑥+4) 3𝑥+8 = 5𝑥 3𝑥+8 ;𝑥≠−4, 0

Monitoring Progress Find the sum or difference: 1) 8 12𝑥 − 5 12𝑥 2) 2 3 𝑥 2 + 1 3 𝑥 2 3) 4𝑥 𝑥−2 − 𝑥 𝑥−2 4) 2 𝑥 2 𝑥 2 +1 + 2 𝑥 2 +1 Find the least common multiple: 5) 5 𝑥 3 ,10 𝑥 2 −15𝑥 6) 3 4𝑥 − 1 7 7) 1 3𝑥 2 + 𝑥 9𝑥 2 −12 8) 𝑥 𝑥 2 −𝑥−12 + 5 12𝑥−48 = 1 4𝑥 = 1 𝑥 2 = 3𝑥 𝑥−2 =2 𝐿𝐶𝑀=5 𝑥 3 (2𝑥−3) = 21−4𝑥 28𝑥 = (𝑥−1) (𝑥+2) 2 3 𝑥 2 (3 𝑥 2 −4) = 17𝑥+15 12(𝑥−4) (𝑥+3)

Monitoring Progress Simplify the complex fractions. 10) 11) 12) 𝑥 6 − 𝑥 3 𝑥 5 − 7 10 = −5𝑥 3(2𝑥−7) 2 𝑥 −4 2 𝑥 +3 = 2(1−2𝑥) 2+3𝑥 3 𝑥+5 2 𝑥−3 + 1 𝑥+5 = 3(𝑥−3) 3𝑥+7