Rational Expressions and

Equations

Chapter 6

§ 6.1

Simplifying, Multiplying, and

Dividing

Tobey & Slater, Intermediate Algebra, 5e - Slide #3

Rational Expressions

(2 4)( 5)

xx

(2 4) ( 5)x x or Fractional

algebraic expression

x – 5

A rational expression is an expression of the form where P and Q are polynomials and Q is not 0.

,PQ

A function defined by a rational expression is a rational function.

The domain of a rational function is the set of values that can be used to replace the variable.

Tobey & Slater, Intermediate Algebra, 5e - Slide #4

Simplifying by Factoring

Example: Find the domain of 25 10 .

2 8

x

x x

Set the denominator equal to 0.x2 – 2x – 8 = 0

Factor.(x + 2)(x – 4) = 0

Use the zero factor property.x + 2 = 0 or x – 4 = 0

Solve for x.x = – 2 x = 4

The domain of y = f(x) is all real numbers except – 2 and 4.

Tobey & Slater, Intermediate Algebra, 5e - Slide #5

Basic Rules of Fractions

Basic Rules of Fractions

For any polynomials a, b, or c,

where b and c 0.,ac abc b

Example: Reduce.21 35

21 3 735 5 7

35

Tobey & Slater, Intermediate Algebra, 5e - Slide #6

Simplifying by Factoring

Example: Simplify. 5 10

2xx

5 102

xx

5( 2)2

xx

Factor 5 from the numerator.

5( 2)2

xx

Apply the basic rule of fractions.

5

Tobey & Slater, Intermediate Algebra, 5e - Slide #7

Simplifying by Factoring

Example: Simplify. 8 63 4x

x

8 63 4x

x

2( 4 3)(3 4 )

xx

Factor – 2 from the numerator.

2( 4 3)3 4

xx

Apply the basic rule of fractions.

2

Remember that when a negative number is factored from a polynomial, the sign of each term in the polynomial changes.

Tobey & Slater, Intermediate Algebra, 5e - Slide #8

Simplifying by Factoring

Example: Simplify. 3 2

23 4010 25

x x xx x

3 2

23 4010 25

x x xx x

2

2

( 3 40)10 25

x x xx x

Factor x from the numerator.

( 8)( 5)( 5)( 5)x x x

x x

Factor the numerator.

( 8)( 5)x x

x

( 8)( 5)( 5)( 5)x x x

x x Apply the basic rule of fractions.

Factor the denominator.

Tobey & Slater, Intermediate Algebra, 5e - Slide #9

Multiplying Rational Expressions

For any polynomials a, b, c, and d,

where b and d 0., a c acb d bd

21 7 147.35 3 105

75

21 7.35 3

1

7 1

5

75

Rational expressions may be multiplied and then simplified.

Rational expressions may also first be simplified and then multiplied.

This method is usually easier.

Tobey & Slater, Intermediate Algebra, 5e - Slide #10

Simplifying the Product

Example: Multiply. 2

27 7 20

4 7 42 49x x xx x x

2

27 7 20

4 7 42 49x x xx x x

2

7( 1) ( 5)( 4)4 7( 6 7)

x x xx x x

Factor each numerator and denominator.

7( 1) ( 5)( 4)4 7( 7)( 1)

x x xx x x

Apply the basic rule of fractions.

57

xx

7( 1) ( 5)( 4)4 7( 7)( 1)

x x xx x x

Factor again whenever possible.

Tobey & Slater, Intermediate Algebra, 5e - Slide #11

Dividing Rational Expressions

The definition for division of fractions is

.a c a db d b c

4 2 3 3

x x

x xExample: Divide.

4 23 3

x xx x

4 3

3 2x x

x x

2

2

Invert the second fraction and multiply. This is called the reciprocal.

4 33 2

x xx x

Apply the basic rule of fractions.

Tobey & Slater, Intermediate Algebra, 5e - Slide #12

Simplifying the Quotient

Example: Divide. 2

24 9 (6 9)

4 12 9x x

x x

2

24 9 (6 9)

4 12 9x x

x x

2

24 9 1

(6 9)4 12 9x

xx x

Invert the second fraction and multiply.

(2 3)(2 3) 1(2 3)(2 3) 3(2 3)

x xx x x

Apply the basic rule of fractions.

13(2 3)x

(2 3)(2 3) 1(2 3)(2 3) 3(2 3)

x xx x x

Factor the numerator and denominator.