Addition and Subtraction of Rational Expressions Section 6.3 Addition and Subtraction of Rational Expressions

Objectives Least Common Multiples Review of Addition and Subtraction of Fractions Addition of Rational Expressions Subtraction of Rational Expressions

FINDING THE LEAST COMMON MULTIPLE The least common multiple (LCM) of two or more polynomials can be found as follows. Step 1: Factor each polynomial completely. Step 2: List each factor the greatest number of times that it occurs in either factorization. Step 3: Find the product of this list of factors. The result is the LCM.

Example Find the least common multiple of each pair of expressions. a. 6x, 9x4 b. x2 + 7x + 12, x2 + 8x + 16 Solution Step 1: Factor each polynomial completely. 6x = 3 ∙ 2 ∙ x 9x4 = 3 ∙ 3 ∙ x ∙ x ∙ x ∙ x Step 2: List each factor the greatest number of times. 3 ∙ 3 ∙ 2 ∙ x ∙ x ∙ x ∙ x Step 3: The LCM is 18x4.

Example (cont) b. x2 + 7x + 12, x2 + 8x + 16 Step 1: Factor each polynomial completely. x2 + 7x + 12 = (x + 3)(x + 4) x2 + 8x + 16 = (x+ 4)(x + 4) Step 2: List each factor the greatest number of times. (x + 3), (x + 4), and (x + 4) Step 3: The LCM is (x + 3)(x + 4)2.

Example Find the sum. a. b. Solution a. The LCD is 42. b. The LCD is 18.

Example Find the difference. a. b. Solution a. The LCD is 36. b. The LCD is 60.

SUMS OF RATIONAL EXPRESSIONS To add two rational expressions with like denominators, add their numerators. The denominator does not change. C not zero

Example Add and simplify. a. b. Solution a. b.

Example Add and simplify. a. b. Solution

DIFFERENCES OF RATIONAL EXPRESSIONS To subtract two rational expressions with like denominators, subtract their numerators. The denominator does not change. C not zero

Example Subtract and simplify. a. b. Solution a. b.

Example Subtract and simplify. Solution The LCD is x(x + 7).

Example Simplify the expression. Write your answer in lowest terms and leave it in factored form. Solution

STEPS FOR FINDING SUMS AND DIFFERENCES OF RATIONAL EXPRESSIONS Step 1: If the denominators are not common, multiply each expression by 1 written in the appropriate form to obtain the LCD. Step 2: Add or subtract the numerators. Combine like terms. Step 3: If possible, simplify the final expression.

Example A 75-watt light bulb with a resistance of R1 = 160 ohms and a 60-watt light bulb with a resistance of R2 = 240 ohms are placed in an electrical circuit. Find the combined resistance. Solution R = 96 ohms