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**Warm up 1. 2. Factor the expression. 10x – 5x2 ANSWER 5x (2 – x)**

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**3. 4. Lesson 8.4, For use with pages 573-580 Factor the expression.**

ANSWER (x – 5)(x2 + 5x + 25) 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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**Add numerators and simplify result.**

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

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**Subtract numerators and simplify results .**

GUIDED PRACTICE for Example 1 Perform the indicated operation and simplify. a. 7 12x + 5 = 7 – 5 12x = 2 12x = 1 6x Subtract numerators and simplify results . b. 2 3x2 + 1 = 2 + 1 3x2 = 3 3x2 = 1 x2 Add numerators and simplify results. c. 4x x–2 – x = 4x–x x–2 = 3x x–2 = 3x 3x – 2 Subtract numerators.

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EXAMPLE 2 Find a least common multiple (LCM) Find the least common multiple of 4x2 –16 and 6x2 –24x + 24. Factor each polynomial. Write numerical factors as products of primes.

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EXAMPLE 3 Add with unlike denominators Add: 9x2 7 + x 3x2 + 3x SOLUTION

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EXAMPLE 4 Subtract with unlike denominators Subtract: x + 2 2x – 2 –2x –1 x2 – 4x + 3 – SOLUTION

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GUIDED PRACTICE for Examples 2, 3 and 4 Find the least common multiple of the polynomials. 5. 5x3 and 10x2–15x

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GUIDED PRACTICE for Examples 2, 3 and 4 Find the least common multiple of the polynomials. 6. 8x – 16 and 12x2 + 12x – 72

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

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GUIDED PRACTICE for Examples 2, 3 and 4 1 3x2 + x 9x2 – 12x 8.

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GUIDED PRACTICE for Examples 2, 3 and 4 x x2 – x – 12 + 5 12x – 48 9. SOLUTION

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**Add numerators Simplify GUIDED PRACTICE for Examples 2, 3 and 4**

12x + 5x + 15 12(x + 3)(x – 4) = Add numerators 17x + 15 12(x +3)(x + 4) = Simplify

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GUIDED PRACTICE for Examples 2, 3 and 4 x + 1 x2 + 4x + 4 – 6 x2 – 4 10. SOLUTION

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EXAMPLE 5 Simplify a complex fraction (Method 1) Physics 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. Lens equation: f 1 p q + =

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**Multiply numberator and denominator by the LCD**

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

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

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

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Operations on Rational Expressions. Rational expressions are fractions in which the numerator and denominator are polynomials and the denominator does.

Operations on Rational Expressions. Rational expressions are fractions in which the numerator and denominator are polynomials and the denominator does.

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