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6.2 Rational Expressions and Functions: Adding and Subtracting ■ When Denominators Are the Same ■ When Denominators Are Different

Slide 6- 3 Copyright © 2012 Pearson Education, Inc. Addition and Subtraction with Like Denominators To add or subtract when the denominators are the same, add or subtract the numerators and keep the same denominator.

Slide 6- 4 Copyright © 2012 Pearson Education, Inc. Example Add. Simplify the result, if possible. a)b) c)d) Solution a) b) The denominators are alike, so we add the numerators.

Slide 6- 5 Copyright © 2012 Pearson Education, Inc. Example continued c) d) Factoring Combining like terms Combining like terms in the numerator

Slide 6- 6 Copyright © 2012 Pearson Education, Inc. Example Subtract and, if possible, simplify: a)b) Solution a) The parentheses are needed to make sure that we subtract both terms. Removing the parentheses and changing the signs (using the distributive law) Combining like terms

Slide 6- 7 Copyright © 2012 Pearson Education, Inc. Example continued b) Removing the parentheses (using the distributive law) Factoring, in hopes of simplifying Removing a factor equal to 1

Slide 6- 8 Copyright © 2012 Pearson Education, Inc. Least Common Multiples and Denominators To add or subtract rational expressions that have different denominators, we must first find equivalent rational expressions that do have a common denominator. The least common multiple (LCM) must include the factors of each number, so it must include each prime factor the greatest number of times that it appears in any factorizations.

Slide 6- 9 Copyright © 2012 Pearson Education, Inc. Example For each pair of polynomials, find the least common multiple. a) 16a and 24b b) 24x 4 y 4 and 6x 6 y 2 c) x 2  4 and x 2  2x  8 Solution a) 16a = 2  2  2  2  a 24b = 2  2  2  3  b The LCM = 2  2  2  2  a  3  b The LCM is 2 4  3  a  b, or 48ab 16a is a factor of the LCM 24b is a factor of the LCM

Slide 6- 10 Copyright © 2012 Pearson Education, Inc. Example continued b) 24x 4 y 4 = 2  2  2  3  x  x  x  x  y  y  y  y 6x 6 y 2 = 2  3  x  x  x  x  x  x  y  y LCM = 2  2  2  3  x  x  x  x  y  y  y  y  x  x Note that we used the highest power of each factor. The LCM is 24x 6 y 4 c) x 2  4 = (x  2)(x + 2) x 2  2x  8 = (x + 2)(x  4) LCM = (x  2)(x + 2)(x  4) x 2  4 is a factor of the LCM x 2  2x  8 is a factor of the LCM

Slide 6- 11 Copyright © 2012 Pearson Education, Inc. Example For each group of polynomials, find the least common multiple. a) 15x, 30y, 25xyzb) x 2 + 3, x + 2, 7 Solution a) 15x = 3  5  x 30y = 2  3  5  y 25xyz = 5  5  x  y  z LCM = 2  3  5  5  x  y  z The LCM is 2  3  5 2  x  y  z or 150xyz b) Since x 2 + 3, x + 2, and 7 are not factorable, the LCM is their product: 7(x 2 + 3)(x + 2).

Slide 6- 12 Copyright © 2012 Pearson Education, Inc. To Add or Subtract Rational Expressions 1. Determine the least common denominator (LCD) by finding the least common multiple of the denominators. 2. Rewrite each of the original rational expressions, as needed, in an equivalent form that has the LCD. 3. Add or subtract the resulting rational expressions, as indicated. 4. Simplify the result, if possible, and list any restrictions, on the domain of functions.

Slide 6- 13 Copyright © 2012 Pearson Education, Inc. Example Add: Solution 1. First, we find the LCD: 9 = 3  3 12 = 2  2  3 2. Multiply each expression by the appropriate number to get the LCD. LCD = 2  2  3  3 = 36

Slide 6- 14 Copyright © 2012 Pearson Education, Inc. Example continued 3. Next we add the numerators: 4. Since 16x 2 + 15x and 36 have no common factor, cannot be simplified any further. Subtraction is performed in much the same way.

Slide 6- 15 Copyright © 2012 Pearson Education, Inc. Example Subtract: Solution We follow the four steps as shown in the previous example. First, we find the LCD. 9x = 3  3  x 12x 2 = 2  2  3  x  x The denominator 9x must be multiplied by 4x to obtain the LCD. The denominator 12x 2 must be multiplied by 3 to obtain the LCD. LCD = 2  2  3  3  x  x = 36x 2

Slide 6- 16 Copyright © 2012 Pearson Education, Inc. Example continued Multiply to obtain the LCD and then we subtract and, if possible, simplify. Caution! Do not simplify these rational expressions or you will lose the LCD. This cannot be simplified, so we are done.

Slide 6- 17 Copyright © 2012 Pearson Education, Inc. Example Add: Solution First, we find the LCD: a 2  4 = (a  2)(a + 2) a 2  2a = a(a  2) We multiply by a form of 1 to get the LCD in each expression: LCD = a(a  2)(a + 2).

Slide 6- 18 Copyright © 2012 Pearson Education, Inc. Example continued 3a 2 + 2a + 4 will not factor so we are done.

Slide 6- 19 Copyright © 2012 Pearson Education, Inc. Example Subtract: Solution First, we find the LCD. It is just the product of the denominators: LCD = (x + 4)(x + 6). We multiply by a form of 1 to get the LCD in each expression. Then we subtract and try to simplify. Multiplying out numerators

Slide 6- 20 Copyright © 2012 Pearson Education, Inc. Example continued Removing parentheses and subtracting every term When subtracting a numerator with more than one term, parentheses are important.