9-5 Adding and Subtracting Rational Expressions Hubarth Algebra II

Ex. 1 Real-world Connection An object is 15cm from a camera lens. The object is in focus on the film when the lens is 10cm from the film. Find the focal length of the lens centimeters. 1 𝑓 = 1 𝑑 𝑖 + 1 𝑑 𝑜 1 𝑓 = 1 10 + 1 15 = 3 30 + 2 30 = 5 30 = 1 6 Since 1 𝑓 = 1 6 , the focal length of the lens is 6cm

Ex. 2 Finding Least Common Multiples Find the least common multiple of 4 𝑥 2 −36 and 6 𝑥 2 +36𝑥+54. 4 𝑥 2 −36=4 𝑥 2 −9 =(2)(2)(𝑥+3)(𝑥−3) 6 𝑥 2 +36𝑥+54=6 𝑥 2 +6𝑥+9 =(3)(2)(x+3)(x+3) (2)(2)(3)(𝑥−3)(𝑥+3 ) 2 =12(x−3)(x+3 ) 2 12(𝑥−3)(𝑥+3 ) 2 is the least common multiple

Ex. 3 Adding Rational Expression Simplify 1 𝑥 2 +5𝑥+4 + 5𝑥 3𝑥+3 1 𝑥 2 +5𝑥+4 + 5𝑥 3𝑥+3 = 1 (𝑥+4)(𝑥+1) + 5𝑥 3(𝑥+1) = 1 (𝑥+4)(𝑥+1) ∙ 3 3 + 5𝑥 3(𝑥+1) ∙ 𝑥+4 𝑥+4 = 3 3(𝑥+4)(𝑥+1) + 5𝑥(𝑥+4) 3(𝑥+1)(𝑥+4) = 3+5𝑥(𝑥+4) 3(𝑥+4)(𝑥+1) = 5 𝑥 2 +20𝑥+3 3(𝑥+4)(𝑥+1)

Ex. 4 Subtracting Rational Expressions Simplify 7𝑦 5 𝑦 2 −125 − 4 3𝑦+15 7𝑦 5 𝑦 2 −125 − 4 3𝑦+15 = 7𝑦 5(𝑦+5)(𝑦−5) − 4 3(𝑦+5) = 7𝑦 5(𝑦+5)(𝑦−5) ∙ 3 3 − 4 3(𝑦+5) ∙ 5(𝑦−5) 5(𝑦−5) = 21𝑦 15(𝑦+5)(𝑦−5) − 20(𝑦−5) 15(𝑦+5)(𝑦−5) = 21𝑦−20𝑦+100 15(𝑦+5)(𝑦−5) = −𝑦+100 15(𝑦+5)(𝑦−5)

A complex fraction is a fraction that has a fraction in its numerator or denominator or in both its numerator and denominator. Ex. 5 Simplifying Complex Fractions Simplify 1 𝑥 +3 5 𝑦 +4 Method 1 Method 2 1 𝑥 +3 5 𝑦 +4 = 1 𝑥 +3 5 𝑦 +4 ∙ 𝑥𝑦 𝑥𝑦 1 𝑥 +3 5 𝑦 +4 = 1 𝑥 + 3𝑥 𝑥 5 𝑦 + 4𝑦 𝑦 = 1+3𝑥 𝑥 5+4𝑦 𝑦 = 1 𝑥 +3∙𝑥𝑦 5 𝑦 +4∙𝑥𝑦 1+3𝑥 𝑥 ÷ 5+4𝑦 𝑦 = 𝑦+3𝑥𝑦 5𝑥+4𝑥𝑦 1+3𝑥 𝑥 ∙ 𝑦 5+4𝑦 = 𝑦+3𝑥𝑦 5𝑥+4𝑥𝑦

Practice Suppose an object is 20cm from a camera lens. When the object is properly focused, the lens is 5cm from the film. Find the focal length of the lens. 4cm 2. Find the least common multiple of the expressions. 5x 2 +15x+10 and 2 x 2 −8 10(𝑥+2)(𝑥−2)(𝑥+1) 3. Simplify 1 𝑥 2 −4𝑥−12 + 3𝑥 4𝑥+8 3 𝑥 2 −18𝑥+4 4(𝑥−6)(𝑥+2) 4. Simplify −2 3 𝑥 2 +36𝑥+105 − 3𝑥 6𝑥+30 −3 𝑥 2 −21𝑥−4 6(𝑥+5)(𝑥+7) 5. Simplify the complex fraction: 𝑥−2 𝑥 − 2 𝑥+1 3 𝑥−1 − 1 𝑥+1 𝑥 3 −4 𝑥 2 +𝑥+2 2 𝑥 2 +4𝑥