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College Algebra

1.6 Rational Expressions

College Algebra1.6 Rational Expressions
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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 7.1 Systems of Linear Equations: Two Variables
    3. 7.2 Systems of Linear Equations: Three Variables
    4. 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 7.4 Partial Fractions
    6. 7.5 Matrices and Matrix Operations
    7. 7.6 Solving Systems with Gaussian Elimination
    8. 7.7 Solving Systems with Inverses
    9. 7.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  9. 8 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 8.1 The Ellipse
    3. 8.2 The Hyperbola
    4. 8.3 The Parabola
    5. 8.4 Rotation of Axes
    6. 8.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  10. 9 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 9.1 Sequences and Their Notations
    3. 9.2 Arithmetic Sequences
    4. 9.3 Geometric Sequences
    5. 9.4 Series and Their Notations
    6. 9.5 Counting Principles
    7. 9.6 Binomial Theorem
    8. 9.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
  12. Index

Learning Objectives

In this section students will:
  • Simplify rational expressions.
  • Multiply rational expressions.
  • Divide rational expressions.
  • Add and subtract rational expressions.
  • Simplify complex rational expressions.

A pastry shop has fixed costs of $280 $280 per week and variable costs of $9 $9 per box of pastries. The shop’s costs per week in terms of x, x, the number of boxes made, is 280+9x. 280+9x. We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.

280+9x x 280+9x x

Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.

Simplifying Rational Expressions

The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.

x 2 +8x+16 x 2 +11x+28 x 2 +8x+16 x 2 +11x+28

We can factor the numerator and denominator to rewrite the expression.

(x+4) 2 (x+4)(x+7) (x+4) 2 (x+4)(x+7)

Then we can simplify that expression by canceling the common factor ( x+4 ). ( x+4 ).

x+4 x+7 x+4 x+7

How To

Given a rational expression, simplify it.

  1. Factor the numerator and denominator.
  2. Cancel any common factors.

Example 1

Simplifying Rational Expressions

Simplify x 2 9 x 2 +4x+3 . x 2 9 x 2 +4x+3 .

Analysis

We can cancel the common factor because any expression divided by itself is equal to 1.

Q&A

Can the x 2 x 2 term be cancelled in Example 1?

No. A factor is an expression that is multiplied by another expression. The x 2 x 2 term is not a factor of the numerator or the denominator.

Try It #1

Simplify x6 x 2 36 . x6 x 2 36 .

Multiplying Rational Expressions

Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.

How To

Given two rational expressions, multiply them.

  1. Factor the numerator and denominator.
  2. Multiply the numerators.
  3. Multiply the denominators.
  4. Simplify.

Example 2

Multiplying Rational Expressions

Multiply the rational expressions and show the product in simplest form:

x2+4x5 3x+18 2x1 x+5 x2+4x5 3x+18 2x1 x+5
Try It #2

Multiply the rational expressions and show the product in simplest form:

x 2 +11x+30 x 2 +5x+6 x 2 +7x+12 x 2 +8x+16 x 2 +11x+30 x 2 +5x+6 x 2 +7x+12 x 2 +8x+16

Dividing Rational Expressions

Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite 1 x ÷ x 2 3 1 x ÷ x 2 3 as the product 1 x 3 x 2 . 1 x 3 x 2 . Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.

1 x 3 x 2 = 3 x 3 1 x 3 x 2 = 3 x 3

How To

Given two rational expressions, divide them.

  1. Rewrite as the first rational expression multiplied by the reciprocal of the second.
  2. Factor the numerators and denominators.
  3. Multiply the numerators.
  4. Multiply the denominators.
  5. Simplify.

Example 3

Dividing Rational Expressions

Divide the rational expressions and express the quotient in simplest form:

2 x 2 +x6 x 2 1 ÷ x 2 4 x 2 +2x+1 2 x 2 +x6 x 2 1 ÷ x 2 4 x 2 +2x+1
Try It #3

Divide the rational expressions and express the quotient in simplest form:

9 x 2 16 3 x 2 +17x28 ÷ 3 x 2 2x8 x 2 +5x14 9 x 2 16 3 x 2 +17x28 ÷ 3 x 2 2x8 x 2 +5x14

Adding and Subtracting Rational Expressions

Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.

5 24 + 1 40 = 25 120 + 3 120 = 28 120 = 7 30 5 24 + 1 40 = 25 120 + 3 120 = 28 120 = 7 30

We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.

The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were (x+3)(x+4) (x+3)(x+4) and (x+4)(x+5), (x+4)(x+5), then the LCD would be (x+3)(x+4)(x+5). (x+3)(x+4)(x+5).

Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of (x+3)(x+4) (x+3)(x+4) by x+5 x+5 x+5 x+5 and the expression with a denominator of (x+4)(x+5) (x+4)(x+5) by x+3 x+3 . x+3 x+3 .

How To

Given two rational expressions, add or subtract them.

  1. Factor the numerator and denominator.
  2. Find the LCD of the expressions.
  3. Multiply the expressions by a form of 1 that changes the denominators to the LCD.
  4. Add or subtract the numerators.
  5. Simplify.

Example 4

Adding Rational Expressions

Add the rational expressions:

5 x + 6 y 5 x + 6 y

Analysis

Multiplying by y y y y or x x x x does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.

Example 5

Subtracting Rational Expressions

Subtract the rational expressions:

6 x 2 +4x+4 2 x 2 −4 6 x 2 +4x+4 2 x 2 −4

Q&A

Do we have to use the LCD to add or subtract rational expressions?

No. Any common denominator will work, but it is easiest to use the LCD.

Try It #4

Subtract the rational expressions: 3 x+5 1 x−3 . 3 x+5 1 x−3 .

Simplifying Complex Rational Expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression a 1 b +c a 1 b +c can be simplified by rewriting the numerator as the fraction a 1 a 1 and combining the expressions in the denominator as 1+bc b . 1+bc b . We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get a 1 b 1+bc , a 1 b 1+bc , which is equal to ab 1+bc . ab 1+bc .

How To

Given a complex rational expression, simplify it.

  1. Combine the expressions in the numerator into a single rational expression by adding or subtracting.
  2. Combine the expressions in the denominator into a single rational expression by adding or subtracting.
  3. Rewrite as the numerator divided by the denominator.
  4. Rewrite as multiplication.
  5. Multiply.
  6. Simplify.

Example 6

Simplifying Complex Rational Expressions

Simplify: y+ 1 x x y y+ 1 x x y .

Try It #5

Simplify: x y y x y x y y x y

Q&A

Can a complex rational expression always be simplified?

Yes. We can always rewrite a complex rational expression as a simplified rational expression.

Media

Access these online resources for additional instruction and practice with rational expressions.

1.6 Section Exercises

Verbal

1.

How can you use factoring to simplify rational expressions?

2.

How do you use the LCD to combine two rational expressions?

3.

Tell whether the following statement is true or false and explain why: You only need to find the LCD when adding or subtracting rational expressions.

Algebraic

For the following exercises, simplify the rational expressions.

4.

x 2 16 x 2 5x+4 x 2 16 x 2 5x+4

5.

y 2 +10y+25 y 2 +11y+30 y 2 +10y+25 y 2 +11y+30

6.

6 a 2 24a+24 6 a 2 24 6 a 2 24a+24 6 a 2 24

7.

9 b 2 +18b+9 3b+3 9 b 2 +18b+9 3b+3

8.

m12 m 2 144 m12 m 2 144

9.

2 x 2 +7x4 4 x 2 +2x2 2 x 2 +7x4 4 x 2 +2x2

10.

6 x 2 +5x4 3 x 2 +19x+20 6 x 2 +5x4 3 x 2 +19x+20

11.

a 2 +9a+18 a 2 +3a18 a 2 +9a+18 a 2 +3a18

12.

3 c 2 +25c18 3 c 2 23c+14 3 c 2 +25c18 3 c 2 23c+14

13.

12 n 2 29n8 28 n 2 5n3 12 n 2 29n8 28 n 2 5n3

For the following exercises, multiply the rational expressions and express the product in simplest form.

14.

x 2 x6 2 x 2 +x6 2 x 2 +7x15 x 2 9 x 2 x6 2 x 2 +x6 2 x 2 +7x15 x 2 9

15.

c 2 +2c24 c 2 +12c+36 c 2 10c+24 c 2 8c+16 c 2 +2c24 c 2 +12c+36 c 2 10c+24 c 2 8c+16

16.

2 d 2 +9d35 d 2 +10d+21 3 d 2 +2d21 3 d 2 +14d49 2 d 2 +9d35 d 2 +10d+21 3 d 2 +2d21 3 d 2 +14d49

17.

10 h 2 9h9 2 h 2 19h+24 h 2 16h+64 5 h 2 37h24 10 h 2 9h9 2 h 2 19h+24 h 2 16h+64 5 h 2 37h24

18.

6 b 2 +13b+6 4 b 2 9 6 b 2 +31b30 18 b 2 3b10 6 b 2 +13b+6 4 b 2 9 6 b 2 +31b30 18 b 2 3b10

19.

2 d 2 +15d+25 4 d 2 25 2 d 2 15d+25 25 d 2 1 2 d 2 +15d+25 4 d 2 25 2 d 2 15d+25 25 d 2 1

20.

6 x 2 5x50 15 x 2 44x20 20 x 2 7x6 2 x 2 +9x+10 6 x 2 5x50 15 x 2 44x20 20 x 2 7x6 2 x 2 +9x+10

21.

t 2 1 t 2 +4t+3 t 2 +2t15 t 2 4t+3 t 2 1 t 2 +4t+3 t 2 +2t15 t 2 4t+3

22.

2 n 2 n15 6 n 2 +13n5 12 n 2 13n+3 4 n 2 15n+9 2 n 2 n15 6 n 2 +13n5 12 n 2 13n+3 4 n 2 15n+9

23.

36 x 2 25 6 x 2 +65x+50 3 x 2 +32x+20 18 x 2 +27x+10 36 x 2 25 6 x 2 +65x+50 3 x 2 +32x+20 18 x 2 +27x+10

For the following exercises, divide the rational expressions.

24.

3 y 2 7y6 2 y 2 3y9 ÷ y 2 +y2 2 y 2 +y3 3 y 2 7y6 2 y 2 3y9 ÷ y 2 +y2 2 y 2 +y3

25.

6 p 2 +p12 8 p 2 +18p+9 ÷ 6 p 2 11p+4 2 p 2 +11p6 6 p 2 +p12 8 p 2 +18p+9 ÷ 6 p 2 11p+4 2 p 2 +11p6

26.

q 2 9 q 2 +6q+9 ÷ q 2 2q3 q 2 +2q3 q 2 9 q 2 +6q+9 ÷ q 2 2q3 q 2 +2q3

27.

18 d 2 +77d18 27 d 2 15d+2 ÷ 3 d 2 +29d44 9 d 2 15d+4 18 d 2 +77d18 27 d 2 15d+2 ÷ 3 d 2 +29d44 9 d 2 15d+4

28.

16 x 2 +18x55 32 x 2 36x11 ÷ 2 x 2 +17x+30 4 x 2 +25x+6 16 x 2 +18x55 32 x 2 36x11 ÷ 2 x 2 +17x+30 4 x 2 +25x+6

29.

144 b 2 25 72 b 2 6b10 ÷ 18 b 2 21b+5 36 b 2 18b10 144 b 2 25 72 b 2 6b10 ÷ 18 b 2 21b+5 36 b 2 18b10

30.

16 a 2 24a+9 4 a 2 +17a15 ÷ 16 a 2 9 4 a 2 +11a+6 16 a 2 24a+9 4 a 2 +17a15 ÷ 16 a 2 9 4 a 2 +11a+6

31.

22 y 2 +59y+10 12 y 2 +28y5 ÷ 11 y 2 +46y+8 24 y 2 10y+1 22 y 2 +59y+10 12 y 2 +28y5 ÷ 11 y 2 +46y+8 24 y 2 10y+1

32.

9 x 2 +3x20 3 x 2 7x+4 ÷ 6 x 2 +4x10 x 2 2x+1 9 x 2 +3x20 3 x 2 7x+4 ÷ 6 x 2 +4x10 x 2 2x+1

For the following exercises, add and subtract the rational expressions, and then simplify.

33.

4 x + 10 y 4 x + 10 y

34.

12 2q 6 3p 12 2q 6 3p

35.

4 a+1 + 5 a3 4 a+1 + 5 a3

36.

c+2 3 c4 4 c+2 3 c4 4

37.

y+3 y2 + y3 y+1 y+3 y2 + y3 y+1

38.

x1 x+1 2x+3 2x+1 x1 x+1 2x+3 2x+1

39.

3z z+1 + 2z+5 z2 3z z+1 + 2z+5 z2

40.

4p p+1 p+1 4p 4p p+1 p+1 4p

41.

x x+1 + y y+1 x x+1 + y y+1

For the following exercises, simplify the rational expression.

42.

6 y 4 x y 6 y 4 x y

43.

2 a + 7 b b 2 a + 7 b b

44.

x 4 p 8 p x 4 p 8 p

45.

3 a + b 6 2b 3a 3 a + b 6 2b 3a

46.

3 x+1 + 2 x1 x1 x+1 3 x+1 + 2 x1 x1 x+1

47.

a b b a a+b ab a b b a a+b ab

48.

2x 3 + 4x 7 x 2 2x 3 + 4x 7 x 2

49.

2c c+2 + c1 c+1 2c+1 c+1 2c c+2 + c1 c+1 2c+1 c+1

50.

x y y x x y + y x x y y x x y + y x

Real-World Applications

51.

Brenda is placing tile on her bathroom floor. The area of the floor is 15 x 2 8x7 15 x 2 8x7 ft2. The area of one tile is x 2 2x+1 ft 2 . x 2 2x+1 ft 2 .To find the number of tiles needed, simplify the rational expression: 15 x 2 8x7 x 2 2x+1 . 15 x 2 8x7 x 2 2x+1 .

A rectangle that’s labeled: Area = fifteen times x squared minus eight times x minus seven.
52.

The area of Sandy’s yard is 25 x 2 625 25 x 2 625 ft2. A patch of sod has an area of x 2 10x+25 x 2 10x+25 ft2. Divide the two areas and simplify to find how many pieces of sod Sandy needs to cover her yard.

53.

Aaron wants to mulch his garden. His garden is x 2 +18x+81 x 2 +18x+81 ft2. One bag of mulch covers x 2 81 x 2 81 ft2. Divide the expressions and simplify to find how many bags of mulch Aaron needs to mulch his garden.

Extensions

For the following exercises, perform the given operations and simplify.

54.

x 2 +x6 x 2 2x3 2 x 2 3x9 x 2 x2 ÷ 10 x 2 +27x+18 x 2 +2x+1 x 2 +x6 x 2 2x3 2 x 2 3x9 x 2 x2 ÷ 10 x 2 +27x+18 x 2 +2x+1

55.

3 y 2 10y+3 3 y 2 +5y2 2 y 2 3y20 2 y 2 y15 y4 3 y 2 10y+3 3 y 2 +5y2 2 y 2 3y20 2 y 2 y15 y4

56.

4a+1 2a3 + 2a3 2a+3 4 a 2 +9 a 4a+1 2a3 + 2a3 2a+3 4 a 2 +9 a

57.

x 2 +7x+12 x 2 +x6 ÷ 3 x 2 +19x+28 8 x 2 4x24 ÷ 2 x 2 +x3 3 x 2 +4x7 x 2 +7x+12 x 2 +x6 ÷ 3 x 2 +19x+28 8 x 2 4x24 ÷ 2 x 2 +x3 3 x 2 +4x7

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