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8.2: Rational Expressions

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Rational Expressions

In arithmetic, it is noted that a fraction is a quotient of two whole numbers. The expression ab, where a and b are any two whole numbers and b0, is called a fraction. The top number, a, is called the numerator, and the bottom number, b, is called the denominator.

Simple Algebraic Fraction

We define a simple algebraic fraction in a similar manner. Rather than restrict­ing ourselves only to numbers, we use polynomials for the numerator and denomi­nator. Another term for a simple algebraic fraction is a rational expression. A rational expression is an expression of the form PQ, where P and Q are both polyno­mials and Q never represents the zero polynomial.

Rational Expression

A rational expression is an algebraic expression that can be written as the quotient of two polynomials.

Examples 1–4 are rational expressions:

Example 8.2.1

x+9x7 is a rational expression: P is x+9 and Q is x7.

Example 8.2.2

x3+5x212x+1x410 is a rational expression. P is x3+5x212x+1 and Q is x410

Example 8.2.3

38 is a rational expression: P is 3 and Q is 8.

Example 8.2.4

4x5 is a rational expression since 4x5 can be written as 4x51: P is 4x5 and Q is 1.

Example 8.2.5

5x282x1 is not a rational expression since 5x28 is not a polynomial.

In the rational expression PQ, P is called the numerator and Q is called the denominator.

Domain of a Rational Expression

Since division by zero is not defined, we must be careful to note the values for which the rational expression is valid. The collection of values for which the rational expression is defined is called the domain of the rational expression. (Recall our study of the domain of an equation in Section 4.8.)

Finding the Domain of a Rational Expression

To find the domain of a rational expression we must ask, "What values, if any, of the variable will make the denominator zero?" To find these values, we set the denominator equal to zero and solve. If any zero-producing values are obtained, they are not included in the domain. All other real numbers are included in the domain (unless some have been excluded for particular situational reasons).

Zero-Factor Property

Sometimes to find the domain of a rational expression, it is necessary to factor the denominator and use the zero-factor property of real numbers.

Zero-Factor Property

If two real numbers a and b are multiplied together and the resulting product is 0, then at least one of the factors must be zero, that is, either a=0,b=0, or both a=0 and b=0.

The following examples illustrate the use of the zero-factor property.

Example 8.2.6

What value will produce zero in the expression 4x? By the zero-factor property, if 4x=0, then x=0.

Example 8.2.7

What value will produce zero in the expression 8(x6)? By the zero-factor property, if 8(x6)=0, then:

x6=0x=0

Thus, 8(x6)=0 when x=6.

Example 8.2.8

What value(s) will produce zero in the expression (x3)(x+5)? By the zero-factor property, if (x3)(x+5)=0, then:

x3=0 or x+5=0x=3x=5

Thus, (x3)(x+5)=0 when x=3 or x=5.

Example 8.2.9

What value(s) will produce zero in the expression x2+6x+8? We must factor x2+6x+8 to put it into the zero-factor property form.

x2+6x+8=(x+2)(x+4)

Now, (x+2)(x+4)=0 when

x+2=0 or x+4=0x=2x=4

Thus, x2+6x+8=0 when x=2 or x=4.

Example 8.2.10

What value(s) will produce zero in the expression 6x219x7? We must factor 6x219x7 to put it into the zero-factor property form.

6x219x7=(3x+1)(2x7)

Now, (3x+1)(2x7)=0 when

3x+1=0 or 2x7=03x=12x=7

Thus, 6x219x7=0 when x=13 or 72

Sample Set A

Find the domain of the following expressions.

Example 8.2.11

5x1

The domain is the collection of all real numbers except 1. One is not included, for if x=1, division by zero results.

Example 8.2.12

3a2a8

If we set 2a8 equal to zero, we find that a=4.

2a8=02a=8a=4

Thus 4 must be excluded from the domain since it will produce division by zero. The domain is the collection of all real numbers except 4.

Example 8.2.13

5x1(x+2)(x6).

Setting (x+2)(x6)=0, we find that x=2 and x=6. Both these values produce division by zero and must be excluded from the domain. The domain is the collection of all real numbers except 2 and 6.

Example 8.2.14

9(x22x15.

Setting x22x15=0, we get:

(x+3)(x5)=0x=3,5

Thus, x=3 and x=5 produce division by zero and must be excluded from the domain. The domain is the collection of all real numbers except 3 and 5.

Example 8.2.15

2x2+x7x(x1)(x3)(x+10)

Setting x(x1)(x3)(x+10)=0, we get x=0,1,3,10. These numbers must be excluded from the domain. The domain is the collection of all real numbers except 0,1,3,10.

Example 8.2.16

8b+7(2b+1)(3b2).

Setting (2b+1)(3b2)=0, we get b=12,23. The domain is the collection of all real numbers except 12 and 23.

Example 8.2.17

4x5x2+1.

No value of x is excluded since for any choice of x, the denominator is never zero. The domain is the collection of all real numbers.

Example 8.2.18

x96

No value of x is excluded since for any choice of x, the denominator is never zero. The domain is the collection of all real numbers.

Practice Set A

Find the domain of each of the following rational expressions.

Practice Problem 8.2.1

2x7

Answer

7

Practice Problem 8.2.2

5xx(x+4)

Answer

0,4

Practice Problem 8.2.3

2x+1(x+2)(1x)

Answer

2,1

Practice Problem 8.2.4

5a+2a2+6a+8

Answer

2,4

Practice Problem 8.2.5

12y3y22y8

Answer

(43,2)

Practice Problem 8.2.6

2m5m2+3

Answer

All real numbers comprise the domain.

Practice Problem 8.2.7

k245

Answer

All real numbers comprise the domain.

The Equality Property of Fractions

From our experience with arithmetic, we may recall the equality property of fractions. Let a,b,c,d be real numbers such that b0 and d0.

Equality Property of Fractions

If ab=cd, then ad=bc.

If ad=bc, then ab=cd

Two fractions are equal when their cross-products are equal.

We see this property in the following examples:

Example 8.2.18

23=812, since 212 =38.

Example 8.2.19

5y2=15y26y, since 5y6y=215y2 and 30y2=30y2.

Example 8.2.20

Since 9a4=18a2, 9a18a=24

The Negative Property of Fractions

A useful property of fractions is the negative property of fractions.

Negative Property of Fractions

The negative sign of a fraction may be placed:

- in front of the fraction, ab,

- in the numerator of the fraction, ab,

- in the denominator of the fraction, ab,

All three fractions will have the same value, that is,

ab=ab=ab

The negative property of fractions is illustrated by the fractions

34=34=34

To see this, consider 34=34. Is this correct?

By the equality property of fractions, (34)=13 and 34=12. Thus, 34=34. Convince yourself that the other two fractions are equal as well.

This same property holds for rational expressions and negative signs. This property is often quite helpful in simplifying a rational expression (as we shall need to do in subsequent sections).

If either the numerator or denominator of a fraction or a fraction itself is immediately preceded by a negative sign, it is usually most convenient to place the negative sign in the numerator for later operations.

Sample Set B

Example 8.2.21

x4 is best written as x4

Example 8.2.21

y9 is best written as y9

Example 8.2.21

x42x5 could be written as (x4)2x5, which would then yield x+42x5

Example 8.2.21

510x. Factor our 1 from the denominator.

5(10+x) A negative divided by a negative is a positive

510+x

Example 8.2.21

37x. Rewrite this.

37x Factor out 1 from the denominator.

3(7+x) A negative divided by a negative is positive.

37+x Rewrite.

3x7

This expression seems less cumbersome than does the original (fewer minus signs).

Practice Set B

Fill in the missing term.

Practice Problem 8.2.8

5y2=?y2

Answer

5

Practice Problem 8.2.9

a+2a+3=?a3

Answer

a+2

Practice Problem 8.2.10

85y=?y5

Answer

8

Exercises

For the following problems, find the domain of each of the rational expressions.

Exercise 8.2.1

6x4

Answer

x4

Exercise 8.2.2

3x8

Exercise 8.2.3

11xx+1

Answer

x1

Exercise 8.2.4

x+10x+4

Exercise 8.2.5

x1x24

Answer

x2,2

Exercise 8.2.6

x+7x29

Exercise 8.2.7

x+4x236

Answer

x6,6

Exercise 8.2.8

a+5a(a5)

Exercise 8.2.9

2bb(b+6)

Answer

b0,6

Exercise 8.2.10

3b+1b(b4)(b+5)

Exercise 8.2.11

3x+4x(x10)(x+1)

Answer

x0,10,1

Exercise 8.2.12

2xx2(4x)

Exercise 8.2.13

6aa3(a5)(7a)

Answer

x0,5,7

Exercise 8.2.14

5a2+6a+8

Exercise 8.2.15

8b24b+3

Answer

b1,3

Exercise 8.2.16

x1x29x+2

Exercise 8.2.17

y9y2y20

Answer

y5,4

Exercise 8.2.18

y62y23y2

Exercise 8.2.19

2x+76x3+x22x

Answer

x0,12,23

Exercise 8.2.20

x+4x38x2+12x

For the following problems, show that the fractions are equivalent.

Exercise 8.2.21

35 and 35

Answer

(3)5=15,(35)=15

Exercise 8.2.22

27 and 27

Exercise 8.2.23

14 and 14

Answer

(14)=4,4(1)=4

Exercise 8.2.24

23 and 23

Exercise 8.2.25

910 and 910

Answer

(9)(10)=90 and (9)(10)=90

For the following problems, fill in the missing term.

Exercise 8.2.26

4x1=?x1

Exercise 8.2.27

2x+7=?x+7

Answer

2

Exercise 8.2.28

3x+42x1=?2x1

Exercise 8.2.29

2x+75x1=?5x1

Answer

2x7

Exercise 8.2.30

x26x1=?6x1

Exercise 8.2.31

x42x3=?2x3

Answer

x+4

Exercise 8.2.32

x+5x3=?x+3

Exercise 8.2.33

a+1a6=?a+6

Answer

a+1

Exercise 8.2.34

x7x+2=?x2

Exercise 8.2.35

y+10y6=?y+6

Answer

y10

Exercises For Review

Exercise 8.2.36

Write (15x3y45x2y7)2 so that only positive exponents appear.

Exercise 8.2.37

Solve the compound inequality 16x5<13

Answer

1x<3

Exercise 8.2.38

Factor 8x218x5.

Exercise 8.2.39

Factor x212x+36

Answer

(x6)2

Exercise 8.2.40

Supply the missing word. The phrase "graphing an equation" is interpreted as meaning "geometrically locate the ____ to an equation."


This page titled 8.2: Rational Expressions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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