Beginning Algebra, 11th Edition

NOTE The numerator of a rational expression may be any real number.If the

numerator equals 0 and the denominator does not equal 0, then the rational expression

equals 0. See Example 1(b).

SECTION 7.1 The Fundamental Property of Rational Expressions 421

Finding Values That Make Rational Expressions Undefined

Find any values of the variable for which each rational expression is undefined.

(a)

We must find any value of xthat makes the denominatorequal to 0 ,

since division by 0 is undefined.

Step 1 Set the denominator equal to 0.

Step 2 Solve. Subtract 2.

Divide by 3.

Step 3 The given expression is undefined for so

(b)

The denominator when xis 3. The given expression is

undefined for 3, so

(c)

Set the denominator equal to 0.

Factor.

Zero-factor property

Solve for m.

The given expression is undefined for 2 and 3, so ,

(d)

2 r

r^2 + 1

mZ 2 mZ3.

m= 2 or m= 3

m- 2 = 0 or m- 3 = 0

1 m - 221 m- 32 = 0

m^2 - 5 m+ 6 = 0

9 m^2

m^2 - 5 m+ 6

xZ3.

8 x^2 + 1 x- 3 = 0

x- 3

-^23 , xZ- 32.

x=-

2

3

3 x=- 2

3 x+ 2 = 0

x+ 5

3 x+ 2

EXAMPLE 2

This denominator will not equal 0 for any value ofr, because is always

greater than or equal to 0, and adding 1 makes the sum greater than or

equal to 1. There are no values for which this expression is undefined.

NOW TRY

r^2

OBJECTIVE 3 Write rational expressions in lowest terms.A fraction such

as is said to be in^23 lowest terms.

Lowest Terms

A rational expression is in lowest termsif the greatest common

factor of its numerator and denominator is 1.

PQ 1 Q Z 02

NOW TRY
EXERCISE 2
Find any values of the vari-
able for which each rational
expression is undefined.

(a)

(b)

(c)

y+ 10

y^2 + 10

2 x

x^2 + 5 x- 14

k- 4

2 k- 1

NOW TRY ANSWERS

2. (a) (b)
   (c)never undefined

kZ^12 xZ-7, xZ 2

Determining When a Rational Expression Is Undefined

Step 1 Set the denominator of the rational expression equal to 0.

Step 2 Solve this equation.

Step 3 The solutions of the equation are the values that make the rational

expression undefined. The variable cannotequal these values.