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Example Add. Simplify the result, if possible. a)b) Solution a) b) Combining like terms Factoring Combining like terms in the numerator

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Example Subtract and, if possible, simplify: a)b) Solution a) The parentheses are needed to make sure that we practice safe math. Removing the parentheses and changing the signs (using the distributive law) Combining like terms

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Example continued b) Removing the parentheses (using the distributive law) Factoring, in hopes of simplifying Removing the clever form of 1

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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

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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

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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).

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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. Example Add: LCD = 2 2 3 3 = 36

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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: Example Add: LCD = a(a 2)(a + 2). 3a 2 + 2a + 4 will not factor so we are done.

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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. Example Subtract: Multiplying out numerators When subtracting a numerator with more than one term, parentheses are important, practice safe math.

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Solution Example Add: Adding numerators

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Continued

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