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Simplifying Rational Expressions

ALGEBRA 1 LESSON 12-3

(For help, go to Lessons 9-5 and 9-6.)

Write each fraction in simplest form.

8 2

25 35

1.

2.

3.

Factor each quadratic expression.

4. x2 x 12 5. x2 6x 8 6. x2 2x 15

7. x2 8x 16 8. x2 x 12 9. x2 7x 12

12-3

Simplifying Rational Expressions

ALGEBRA 1 LESSON 12-3

Solutions

8 2

5 8

1. 8 2 4

2.

4. Factors of 12 with a sum of 1 4 and 3.x2

x 12 (x 4)(x 3)

5. Factors of 8 with a sum of 6 2 and 4.x2

6x 8 (x 2)(x 4)

6. Factors of 15 with a sum of 2 3 and 5.x2

2x 15 (x 3)(x 5)

12-3

Simplifying Rational Expressions

ALGEBRA 1 LESSON 12-3

Solutions (continued)

7. Factors of 16 with a sum of 8 4 and 4.x2

8x 16 (x 4)(x 4) or (x 4)2

8. Factors of 12 with a sum of 1 3 and 4.x2

x 12 (x 3)(x 4)

9. Factors of 12 with a sum of 7 3 and 4.x2

7x 12 (x 3)(x 4)

12-3

Simplifying Rational Expressions

ALGEBRA 1 LESSON 12-3

3x 9 x 3

Simplify .

3 Simplify.

12-3

Simplifying Rational Expressions

ALGEBRA 1 LESSON 12-3

Simplify .

12-3

Simplifying Rational Expressions

ALGEBRA 1 LESSON 12-3

3x 27 81 x2

Simplify .

12-3

Simplifying Rational Expressions

ALGEBRA 1 LESSON 12-3

The baking time for bread depends, in part, on

its size and shape. A good approximation for the

baking time, in minutes, of a cylindrical loaf is

, or , where the radius r

and the length h of the baked loaf are in inches.

Find the baking time for a loaf that is 8 inches

long and has a radius of 3 inches. Round your

answer to the nearest minute.

The baking time is approximately 65 minutes.

12-3

Simplifying Rational Expressions

ALGEBRA 1 LESSON 12-3

Simplify each expression.

3.

1.

2.

6 2x x 3

2

4.

5.

3x 4 4x 1

4x 1 x2

12-3

Multiplying and Dividing Rational Expressions

ALGEBRA 1 LESSON 12-4

(For help, go to Lessons 8-3 and 9-6.)

Simplify each expression. 1. r 2 r 8 2. b3

b4 3. c7 c2 4. 3x4 2x5 5. 5n2 n2 6. 15a3

(3a2)

Factor each polynomial. 7. 2c2 15c 7 8. 15t2

26t 11 9. 2q2 11q 5

12-4

Multiplying and Dividing Rational Expressions

ALGEBRA 1 LESSON 12-4

Solutions

1. r 2 r 8 r (2 8) r 10 2. b3 b4

b(34) b7 3. c7 c2 c(7 2) c5 4.

3x4 2x5 (3 2)(x4 x5) 6x(4 5)

6x9 5. 5n2 n2 5(n2 n2) 5n(2 2)

5n4 6. 15a3(3a2) 15(3)(a3 a2) 45a(3

2) 45a5 7. 2c2 15c 7 (2c 1)(c

7)Check (2c 1)(c 7) 2c2 14c 1c 7

2c2 15c 7 8. 15t2 26t 11 (15t 11)(t

1)Check (15t 11)(t 1) 15t2 15t 11t

11 15t2 26t 11 9. 2q2 11q 5 (2q

1)(q 5)Check (2q 1)(q 5) 2q2 10q

1q 5 2q2 11q 5

12-4

Multiplying and Dividing Rational Expressions

ALGEBRA 1 LESSON 12-4

Multiply.

12-4

Multiplying and Dividing Rational Expressions

ALGEBRA 1 LESSON 12-4

3x 1 4

Multiply and .

12-4

Multiplying and Dividing Rational Expressions

ALGEBRA 1 LESSON 12-4

Multiply and x2 7x 12.

12-4

Multiplying and Dividing Rational Expressions

ALGEBRA 1 LESSON 12-4

Divide by .

12-4

Multiplying and Dividing Rational Expressions

ALGEBRA 1 LESSON 12-4

x2 9x 14 11x

Divide by (8x2 16x).

12-4

Multiplying and Dividing Rational Expressions

ALGEBRA 1 LESSON 12-4

Multiply or divide.

1.

2.

3x 2

3(x 3)

3.

4.

12x

5.

2(x 5) x 1

6.

x 1

12-4

Dividing Polynomials

ALGEBRA 1 LESSON 12-5

(For help, go to Lessons 9-1 and 9-3.)

Write each polynomial in standard form. 1. 9a

4a2 1 2. 3x2 6 5x x3 3. 2

8t Find each product. 4. (2x 4)(x 3) 5.

(3n 4)(n 5) 6. (3a2 1)(2a 7)

12-5

Dividing Polynomials

ALGEBRA 1 LESSON 12-5

Solutions

1. 9a 4a2 1 4a2 9a 1 2. 3x2 6 5x

x3 x3 3x2 5x 6 3. 2 8t 8t 2 4.

(2x 4)(x 3) (2x)(x) (2x)(3) (4)(x)

(4)(3) 2x2 6x 4x 12 2x2 10x 12 5.

(3n 4)(n 5) (3n)(n) (3n)(5)

(4)(n) (4)(5) 3n2 15n 4n 20 3n2

11n 20 6. (3a2 1)(2a 7) (3a2)(2a)

(3a2)(7) (1)(2a) (1)(7) 6a3 21a2 2a

7

12-5

Dividing Polynomials

ALGEBRA 1 LESSON 12-5

Divide (18x3 9x2 15x) by 3x2.

12-5

Dividing Polynomials

ALGEBRA 1 LESSON 12-5

Divide (5x2 2x 3) by (x 2)

12-5

Dividing Polynomials

ALGEBRA 1 LESSON 12-5

(continued)

Step 2 Repeat the process Divide, multiply,

subtract, bring down.

12-5

Dividing Polynomials

ALGEBRA 1 LESSON 12-5

The width and area of a rectangle are shown in

the figure below. What is the length?

The length of the rectangle is (3x2 2x 3) in.

12-5

Dividing Polynomials

ALGEBRA 1 LESSON 12-5

Divide (8x 2 6x2) by (1 x).

Rewrite 8x 2 6x2 as 6x2 8x 2 and 1 x

as x 1. Then divide.

12-5

Dividing Polynomials

ALGEBRA 1 LESSON 12-5

Divide. 1. (x8 x6 x4) x2 2. (4x2 2x 6)

(x 1) 3. (6x3 5x2 11) (2x 3) 4. (29

64x3) (4x 3)

x6 x4 x2

4x 6

12-5

Adding and Subtracting Rational Expressions

ALGEBRA 1 LESSON 12-6

(For help, go to Lessons 1-5 and 906.)

Simplify each expression.

3.

1 2

5 2

Factor each quadratic expression.

12-6

Adding and Subtracting Rational Expressions

ALGEBRA 1 LESSON 12-6

Solutions

6.

5 12

3 4

3 3 4 3

9 12

5 12

5 9 12

4 12

1 3

5 12

12-6

Adding and Subtracting Rational Expressions

ALGEBRA 1 LESSON 12-6

Solutions (continued)

10. Factors of 2 with a sum of 3 1 and 2.x2

3x 2 (x 1)(x 2)

11. Factors of 12 with a sum of 7 3 and 4.y2

7y 12 (y 3)(y 4)

12. Factors of 4 with a sum of 4 2 and 2.t2

4t 4 (t 2)(t 2) or (t 2)2

12-6

Adding and Subtracting Rational Expressions

ALGEBRA 1 LESSON 12-6

Add and .

12-6

Adding and Subtracting Rational Expressions

ALGEBRA 1 LESSON 12-6

Subtract from .

12-6

Adding and Subtracting Rational Expressions

ALGEBRA 1 LESSON 12-6

1 8

Add .

Step 2 Rewrite using the LCD and add.

12-6

Adding and Subtracting Rational Expressions

ALGEBRA 1 LESSON 12-6

Add and .

Step 1 Find the LCD of x 4 and x 5. Since

there are no common factors, the LCD is (x

4)(x 5).

Step 2 Rewrite using the LCD and add.

12-6

Adding and Subtracting Rational Expressions

ALGEBRA 1 LESSON 12-6

The distance between Seattle, Washington, and

Miami, Florida, is about 5415 miles. The ground

speed for jet traffic from Seattle to Miami can

be about 14 faster than the ground speed from

Miami to Seattle. Use r for the jets ground

speed. Write and simplify an expression for the

round-trip air time.

12-6

Adding and Subtracting Rational Expressions

ALGEBRA 1 LESSON 12-6

(continued)

5415 r

5415 1.14r

An expression for the total time is

12-6

Adding and Subtracting Rational Expressions

ALGEBRA 1 LESSON 12-6

18 5x 45x

4(y 1) y 2

9x 20 x 3

12-6

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

(For help, go to Lessons 4-1 and 12-6.)

Solve each proportion.

Find the LCD of each group of expressions.

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

Solutions

1.

2.

3.

m2 81

m 81 9

4. 4n 2 2 n 2 2 n n LCD 2 2 n

4n

5. 3x 3 x 5 5 3x 3 x LCD 3 5

x 15x

6. 8y 2 2 2 y y2 y y 6 2 3

LCD 2 2 2 3 y y 24y2

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

3 8

4 x

14 2x

Solve .

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

(continued)

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

5 x

Solve 1. Check the solution.

(x 3)(x 2) 0 Factor the quadratic

expression.

x 3 0 or x 2 0 Use the Zero-Product

Property.

x 3 or x 2 Solve.

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

(continued)

Check

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

Renee can mow the lawn in 20 minutes. Joanne can

do the same job in 30 minutes. How long will it

take them if they work together?

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

(continued)

Relate Renees part done Joannes part done

complete job.

3n 2n 60 Use the Distributive Property.

It will take two of them 12 minutes to mow the

lawn working together.

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

Solve . Check the solution.

4x 32 x2 Use the Distributive Property.

x2 4x 32 0 Collect terms on one side.

(x 8)(x 4) 0 Factor the quadratic

expression.

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

(continued)

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

Solve .

x2 x 3x 3 4x 4 Use the Distributive

Property.

x2 2x 3 4x 4 Combine like terms.

x2 2x 1 0 Subtract 4x 4 from each side.

(x 1) (x 1) 0 Factor.

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

(continued)

The equation has no solution because 1 makes a

denominator equal 0.

12-7

Solving Rational Equations

ALGEBRA 1 LESSON 12-7

2

1

4

1, 3

12-7

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