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Warm up Factor the expression. ANSWER 5x (2 – x) ANSWER (x + 6)(x – 8) 1. 10x – 5x 2 2. x 2 – 2x – 48

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Lesson 8.4, For use with pages 573-580 3. x 3 – 125 ANSWER (x – 5)(x 2 + 5x + 25) Factor the expression. 4. What is the volume and surface area of a cardboard shipping carton that measures 15 inches by 18 inches by 20 inches? ANSWER 5400 in. 3, 1860 in. 2

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8.5 – Adding and Subtracting Rational Expressions

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EXAMPLE 1 Add or subtract with like denominators Perform the indicated operation. 7 4x4x + 3 4x4x a. 2x2x x + 6 – 5 b. SOLUTION 7 4x4x + 3 4x4x a. = 7 + 3 4x4x 10 4x4x = 5 2x2x = Add numerators and simplify result. x + 6 2x – 5 = 2x2x x + 6 5 – b. Subtract numerators.

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GUIDED PRACTICE for Example 1 Perform the indicated operation and simplify. 1 6x6x a. 7 12x + 5 = 7 – 5 12x = 2 = Subtract numerators and simplify results. 1 x 2 b. 2 3x 2 + 1 3x23x2 = 2 + 1 3x23x2 = 3 3x23x2 = Add numerators and simplify results. c. 4x x–2 – x = 4x–x4x–x = 3x x–2 = Subtract numerators. 3x 3x – 2

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EXAMPLE 2 Find a least common multiple (LCM) Find the least common multiple of 4x 2 –16 and 6x 2 –24x + 24. Factor each polynomial. Write numerical factors as products of primes.

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EXAMPLE 3 Add with unlike denominators Add: 9x29x2 7 + x 3x 2 + 3x SOLUTION

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EXAMPLE 4 Subtract with unlike denominators Subtract: x + 2 2x – 2 –2x –1 x 2 – 4x + 3 – SOLUTION

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GUIDED PRACTICE for Examples 2, 3 and 4 Find the least common multiple of the polynomials. 5. 5x 3 and 10x 2 –15x

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GUIDED PRACTICE for Examples 2, 3 and 4 Find the least common multiple of the polynomials. 6. 8x – 16 and 12x 2 + 12x – 72

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GUIDED PRACTICE for Examples 2, 3 and 4 4x4x 3 – 7 1 7. SOLUTION

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GUIDED PRACTICE for Examples 2, 3 and 4 1 3x23x2 + x 9x 2 – 12x 8.

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GUIDED PRACTICE for Examples 2, 3 and 4 x x 2 – x – 12 + 5 12x – 48 9. SOLUTION

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GUIDED PRACTICE for Examples 2, 3 and 4 12x + 5x + 15 12(x + 3)(x – 4) = 17x + 15 12(x +3)(x + 4) = Add numerators Simplify

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GUIDED PRACTICE for Examples 2, 3 and 4 x + 1 x 2 + 4x + 4 – 6 x 2 – 4 10. SOLUTION

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EXAMPLE 5 Simplify a complex fraction (Method 1) Let f be the focal length of a thin camera lens, p be the distance between an object being photographed and the lens, and q be the distance between the lens and the film. For the photograph to be in focus, the variables should satisfy the lens equation below. Simplify the complex fraction. Physics Lens equation: f 1 1 p 1 q + =

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GUIDED PRACTICE for Examples 5 and 6 x 6 x 3 – x 5 7 10 – – 5x 3 (2x – 7) = Multiply numberator and denominator by the LCD Simplify x 6 x 3 – x 5 7 10 – 11. x 6 x 3 – x 5 7 10 – 30 =

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GUIDED PRACTICE for Examples 5 and 6 2 x – 2 x + 4 3 12.

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GUIDED PRACTICE for Examples 5 and 6 3 x + 5 2 x – 3 + 1 x + 5 13.

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