10-3 Simplifying Rational Expressions Warm Up Warm Up Lesson Presentation Lesson Presentation...

10-3 Simplifying Rational Expressions

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Lesson Presentation

California Standards

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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


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

California Standards


rational expression

Vocabulary


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.


Additional Example 1A: Identifying Excluded Values

Find any excluded values of each rational expression.

g + 4 = 0

g = –4

The excluded value is –4.

Set the denominator equal to 0.

Solve for g by subtracting 4 from each side.


Additional Example 1B: Identifying Excluded Values


x2 – 15x = 0

x(x – 15) = 0

x = 0 or x – 15 = 0

x = 15


Factor.

Use the Zero Product Property.

Solve for x.

The excluded values are 0 and 15.

x = 0 or


Additional Example 1C: Identifying Excluded Values


y2 + 5y + 4 = 0

(y + 4)(y + 1) = 0

y + 4 = 0 or y + 1 = 0

y = –4 or y = –1

The excluded values are –4 and –1.


Factor.


Solve each equation for y.


To review the Zero Product Property, see Lesson 9-5.

To review factoring trinomials, see Chapter 8.

Remember!


Check It Out! Example 1a


t2 + 5 = 0

There are no excluded values.


There are no values of t that make the denominator equal to 0.

10-3 Simplifying Rational ExpressionsCheck It Out! Example 1b


b2 + 5b = 0

b(b + 5) = 0

b = 0 or b + 5 = 0

Set the denominator equal to 0.Factor.


Solve for b.

The excluded values are 0 and –5.

b = –5b = 0 or

10-3 Simplifying Rational ExpressionsCheck It Out! Example 1c


k2 + 7k + 12 = 0

(k + 4)(k + 3) = 0

k + 4 = 0 or k + 3 = 0

k = –4 or k = –3

The excluded values are –4 and –3.


Factor.


Solve each equation for k.


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.


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.

4


Additional Example 2B: Simplifying Rational Expressions


3n; n ≠

Factor 6n² + 3n.

Divide out common factors.

Note that if n = , the

expression is undefined.Simplify. The excluded value

is .


Additional Example 2C: Simplifying Rational Expressions


3p – 2 = 0

3p = 2

There are no common factors. Add 2 to both sides.

Divide both sides by 3. The

excluded value is


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

Caution


Check It Out! Example 2a


Factor 15.

Divide out common factors. Note that if m = 0, the expression is undefined.

Simplify. The excluded value is 0.


Check It Out! Example 2b


Factor the numerator.

Divide out common factors. Note that the expression is not undefined.

Simplify. There is no excluded value.


Check It Out! Example 2c


The numerator and denominator have no common factors. The excluded value is 2.


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


Additional Example 3: Simplifying Rational Expressions with Trinomials

Simplify each rational expression, if possible.

Factor the numeratorand the denominator

when possible.


Simplify.

A. B.


Check It Out! Example 3


Factor the numeratorand the denominator

when possible.


Simplify.

a. b.


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,


Additional Example 4: Simplifying Rational Expressions Using Opposite Binomials


Factor.

Rewrite one opposite binomial.

Identify opposite binomials.

A. B.


Additional Example 4 Continued



Simplify.

A. B.




Factor.

Rewrite one opposite binomial.

Identify opposite binomials.

a. b.


Check It Out! Example 4 Continued



Simplify.

a. b.




Factor.


c.

Simplify.


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 = 4r2.)

Write the ratio of volume to surface area.




To divide by 4 multiply by the reciprocal of 4.


Simplify.

Use properties of exponents.



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.

10-3 Simplifying Rational ExpressionsCheck 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.

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.

Write the ratio of surface to volume twice. Substitute 6 and 3 for r.

Compare the ratios.


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. 9 > 3

Remember!

10-3 Simplifying Rational ExpressionsLesson Quiz: Part I


1. 2. 0 0, 2


3. 4.

5.

10-3 Simplifying Rational ExpressionsLesson 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.

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