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

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

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

8.5 – Adding and Subtracting Rational Expressions

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.

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.

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.

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

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

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

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

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

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

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

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

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

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 + =

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

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

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