Homework: Part I

1. Homework: Part I Find any excluded values of each rational expression. 2. 1. Simplify each rational expression, if possible. 3. 4. 5.

2. Homework: Part II 6. Calvino is building a rectangular tree house. The length is 10 feet longer than the width. His friend Fabio is also building a tree house, but his is square. The sides of Fabio’s tree house are equal to the width of Calvino’s tree house. a. What is the ratio of the area of Calvino’s tree house to the area of Fabio’s tree house? b. Use this ratio to find the ratio of the areas if the width of Calvino’s tree house is 14 feet.

3. Warm Up Simplify each expression. 1. 2. Factor each expression. 3. x2 + 5x + 6 4. 4x2 – 64 (x + 2)(x + 3) 5. 2x2 + 3x + 1 6. 9x2 + 60x + 100 (2x + 1)(x + 1) 4(x + 4)(x – 4) (3x +10)2

4. Vocabulary rational expression

5. A rational expression is an algebraic expression whose numerator and denominator are polynomials.The value of the polynomial expression in the denominator cannot be zero since division by zero is undefined. This means that rational expressions, like rational functions, may have excluded values.

6. Additional Example 1A: Identifying Excluded Values Find any excluded values of each rational expression. g + 4 = 0 Set the denominator equal to 0. g = –4 Solve for g by subtracting 4 from each side. The excluded value is –4.

7. Additional Example 1B: Identifying Excluded Values Find any excluded values of each rational expression. x2 – 15x = 0 Set the denominator equal to 0. Factor. x(x – 15) = 0 x = 0 or x – 15 = 0 Use the Zero Product Property. x = 0 or x = 15 Solve for x. The excluded values are 0 and 15.

8. Additional Example 1C: Identifying Excluded Values Find any excluded values of each rational expression. y2 + 5y + 4 = 0 Set the denominator equal to 0. (y + 4)(y + 1) = 0 Factor. y + 4 = 0 or y + 1 = 0 Use the Zero Product Property. y = –4 or y =–1 Solve each equation for y. The excluded values are –4 and –1.

9. Partner Share! Example 1a Find any excluded values of each rational expression. Set the denominator equal to 0. t2+ 5 = 0 There are no values of t that make the denominator equal to 0. There are no excluded values.

10. b = 0 or b = –5 Partner Share! Example 1b Find any excluded values of each rational expression. b2 + 5b = 0 Set the denominator equal to 0. Factor. b(b + 5) = 0 b = 0 or b + 5 = 0 Use the Zero Product Property. Solve for b. The excluded values are 0 and –5.

11. Partner Share! Example 1c Find any excluded values of each rational expression. k2 + 7k + 12 = 0 Set the denominator equal to 0. (k + 4)(k + 3) = 0 Factor. k + 4 = 0 or k + 3 = 0 Use the Zero Product Property. k = –4 or k =–3 Solve each equation for k. The excluded values are –4 and –3.

12. A rational expression is in its simplest form when the numerator and denominator have no common factors except 1. Remember that to simplify fractions, you can divide out common factors that appear in both the numerator and the denominator. You can do the same to simplify rational expressions.

13. 4 Additional Example 2A: Simplifying Rational Expressions Simplify each rational expression, if possible. Identify any excluded values. Factor 14. Divide out common factors. Note that if r = 0, the expression is undefined. Simplify. The excluded value is 0.

14. Divide out common factors. Note that if n = , the expression is undefined. 3n; n ≠ Simplify. The excluded value is . Additional Example 2B: Simplifying Rational Expressions Simplify each rational expression, if possible. Identify any excluded values. Factor 6n² + 3n.

15. Divide both sides by 3. The excluded value is Additional Example 2C: Simplifying Rational Expressions Simplify each rational expression, if possible. Identify any excluded values. There are no common factors. Add 2 to both sides. 3p – 2 = 0 3p = 2

16. Caution Be sure to use the original denominator when finding excluded values. The excluded values may not be “seen” in the simplified denominator.

17. Partner Share! Example 2a Simplify each rational expression, if possible. Identify any excluded values. Factor 15. Divide out common factors. Note that if m = 0, the expression is undefined. Simplify. The excluded value is 0.

18. Partner Share! Example 2b Simplify each rational expression, if possible. Identify any excluded values. Factor the numerator. Divide out common factors. Note that the expression is not undefined. Simplify. There is no excluded value.

19. Partner Share! Example 2c Simplify each rational expression, if possible. Identify any excluded values. The numerator and denominator have no common factors. The excluded value is 2.

20. From this point forward, you do not need to include excluded values in your answers unless they are asked for.

21. Additional Example 3: Simplifying Rational Expressions with Trinomials Simplify each rational expression, if possible. A. B. Factor the numerator and the denominator when possible. Divide out common factors. Simplify.

22. Partner Share! Example 3 Simplify each rational expression, if possible. a. b. Factor the numerator and the denominator when possible. Divide out common factors. Simplify.

23. Consider The numerator and denominator are opposite binomials. Therefore, Recall from Chapter 8 that opposite binomials can help you factor polynomials. Recognizing opposite binomials can also help you simplify rational expressions.

24. Additional Example 4: Simplifying Rational Expressions Using Opposite Binomials Simplify each rational expression, if possible. A. B. Factor. Identify opposite binomials. Rewrite one opposite binomial.

25. Additional Example 4 Continued Simplify each rational expression, if possible. A. B. Divide out common factors. Simplify.

26. Partner Share! Example 4 Simplify each rational expression, if possible. a. b. Factor. Identify opposite binomials. Rewrite one opposite binomial.

27. Partner Share! Example 4 Continued Simplify each rational expression, if possible. a. b. Divide out common factors. Simplify.

28. Partner Share! Example 4 Simplify each rational expression, if possible. c. Factor. Divide out common factors. Simplify.

29. a. What is the ratio of the theater’s volume to its surface area? (Hint: For a sphere, V = Additional Example 5: Application A theater at an amusement park is shaped like a sphere. The sphere is held up with support rods. and S = 4r2.) Write the ratio of volume to surface area. Divide out common factors.

30. Additional Example 5 Continued Use properties of exponents. To divide by 4 multiply by the reciprocal of 4. Divide out common factors. Simplify.

31. Additional Example 5 Continued b. Use this ratio to find the ratio of the theater’s volume to its surface area when the radius is 45 feet. Write the ratio of volume to surface area. Substitute 45 for r. The ratio of volume to surface area of the theater is 15:1.

32. Lesson Review: Part I Find any excluded values of each rational expression. 0, 2 2. 1. 0 Simplify each rational expression, if possible. 3. 4. 5.

33. Lesson Review: Part II 6. Calvino is building a rectangular tree house. The length is 10 feet longer than the width. His friend Fabio is also building a tree house, but his is square. The sides of Fabio’s tree house are equal to the width of Calvino’s tree house. a. What is the ratio of the area of Calvino’s tree house to the area of Fabio’s tree house? b. Use this ratio to find the ratio of the areas if the width of Calvino’s tree house is 14 feet.