1. Preview
Warm Up
California Standards
Lesson Presentation

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

3. California
Standards
12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

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.
x(x – 15) = 0
Factor.
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 = – or y = –1
Solve each equation for y.
The excluded values are –4 and –1.

9. To review the Zero Product Property,
see Lesson 9-5.
To review factoring trinomials, see Chapter 8.
Remember!

10. Check It Out! 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.

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

12. Check It Out! 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.

13. 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.

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

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

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

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

18. Check It Out! 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.

19. Check It Out! 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.

20. Check It Out! 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.

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

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

23. Divide out common factors.
Check It Out! Example 3
Simplify each rational expression, if possible. a. b.
Factor the numerator
and the denominator
when possible.
Divide out common factors.
Simplify.

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

25. Simplify each rational expression, if possible.
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.

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

27. Simplify each rational expression, if possible.
Check It Out! Example 4
Simplify each rational expression, if possible. a. b.
Factor.
Identify opposite binomials.
Rewrite one opposite binomial.

28. Check It Out! Example 4 Continued
Simplify each rational expression, if possible. a. b.
Divide out common factors.
Simplify.

29. Check It Out! Example 4
Simplify each rational expression, if possible. c.
Factor.
Divide out common factors.
Simplify.

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

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

32. 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.

33. Check It Out! Example 5
Which barrel cactus has less of a chance to survive in the desert, one with a radius of 6 inches or one with a radius of 3 inches? Explain.
Write the ratio of surface to volume twice. Substitute 6 and 3 for r.
Compare the ratios.
The barrel cactus with a radius of 3 inches has less of a chance to survive. Its surface-area-to-volume ratio is greater than for a cactus with a radius of 6 inches.

34. For two fractions with the same numerator, the value of the fraction with a greater denominator is less than the value of the other fraction > 3
Remember!

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

36. Lesson Quiz: 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.