<< Chapter < Page Chapter >> Page >

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.

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

Got questions? Get instant answers now!

What value will produce zero in the expression 8 ( x - 6 ) ? By the zero-factor property, if 8 ( x - 6 ) = 0 , then

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

Got questions? Get instant answers now!

What value(s) will produce zero in the expression ( x - 3 ) ( x + 5 ) ? By the zero-factor property, if ( x - 3 ) ( x + 5 ) = 0 , then

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

Got questions? Get instant answers now!

What value(s) will produce zero in the expression x 2 + 6 x + 8 ? We must factor x 2 + 6 x + 8 to put it into the zero-factor property form.

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

Now, ( x + 2 ) ( x + 4 ) = 0 when
x + 2 = 0 or x + 4 = 0 x = - 2 x = - 4
Thus, x 2 + 6 x + 8 = 0 when x = - 2 or x = - 4 .

Got questions? Get instant answers now!

What value(s) will produce zero in the expression 6 x 2 - 19 x - 7 ? We must factor 6 x 2 - 19 x - 7 to put it into the zero-factor property form.

6 x 2 - 19 x - 7 = ( 3 x + 1 ) ( 2 x - 7 )

Now, ( 3 x + 1 ) ( 2 x - 7 ) = 0 when
3 x + 1 = 0 or 2 x - 7 = 0 3 x = - 1 2 x = 7 x = - 1 3 x = 7 2
Thus, 6 x 2 - 19 x - 7 = 0 when x = - 1 3 or 7 2 .

Got questions? Get instant answers now!

Sample set a

Find the domain of the following expressions.

5 x - 1 .

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

Got questions? Get instant answers now!

3 a 2 a - 8 .

If we set 2 a - 8 equal to zero, we find that a = 4 .

2 a - 8 = 0 2 a = 8 a = 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.

Got questions? Get instant answers now!

5 x - 1 ( x + 2 ) ( x - 6 ) .

Setting ( x + 2 ) ( x - 6 ) = 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.

Got questions? Get instant answers now!

9 x 2 - 2 x - 15 .

Setting x 2 - 2 x - 15 = 0 , we get

( x + 3 ) ( x - 5 ) = 0 x = - 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.

Got questions? Get instant answers now!

2 x 2 + x - 7 x ( x - 1 ) ( x - 3 ) ( x + 10 ) .

Setting x ( x - 1 ) ( x - 3 ) ( 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.

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

4 x - 5 x 2 + 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.

Got questions? Get instant answers now!

x - 9 6 .

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.

Got questions? Get instant answers now!

Practice set a

Find the domain of each of the following rational expressions.

5 x x ( x + 4 )

0 , 4

Got questions? Get instant answers now!

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

2 , 1

Got questions? Get instant answers now!

5 a + 2 a 2 + 6 a + 8

2 , 4

Got questions? Get instant answers now!

12 y 3 y 2 - 2 y - 8

- 4 3 , 2

Got questions? Get instant answers now!

2 m - 5 m 2 + 3

All real numbers comprise the domain.

Got questions? Get instant answers now!

k 2 - 4 5

All real numbers comprise the domain.

Got questions? Get instant answers now!

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 b 0 and d 0 .

Equality property of fractions

  1. If a b = c d , then a d = b c .
  2. If a d = b c , then a b = c d .

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

We see this property in the following examples:

2 3 = 8 12 , since 2 · 12 = 3 · 8 .

Got questions? Get instant answers now!

5 y 2 = 15 y 2 6 y , since 5 y · 6 y = 2 · 15 y 2 and 30 y 2 = 30 y 2 .

Got questions? Get instant answers now!

Since 9 a · 4 = 18 a · 2 , 9 a 18 a = 2 4 .

Got questions? Get instant answers now!

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

  1. in front of the fraction, - a b ,
  2. in the numerator of the fraction, - a b ,
  3. in the denominator of the fraction, a - b .

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

    - a b = - a b = a - b

  • The negative property of fractions is illustrated by the fractions
  • - 3 4 = - 3 4 = 3 - 4

To see this, consider - 3 4 = - 3 4 . Is this correct?

By the equality property of fractions, - 3 · 4 = - 12 and - 3 · 4 = - 12 . Thus, - 3 4 = - 3 4 . 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

x - 4 is best written as - x 4 .

Got questions? Get instant answers now!

- y 9 is best written as - y 9 .

Got questions? Get instant answers now!

- x - 4 2 x - 5 could be written as - x - 4 2 x - 5 , which would then yield - x + 4 2 x - 5 .

Got questions? Get instant answers now!

- 5 - 10 - x . Factor out - 1 from the denominator . - 5 - ( 10 + x ) A negative divided by a negative is a positive . 5 10 + x

Got questions? Get instant answers now!

- 3 7 - x . Rewrite this . - 3 7 - x Factor out - 1 from the deno min ator . - 3 - ( - 7 + x ) A negative divided by a negative is positive . 3 - 7 + x Rewrite . 3 x - 7

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

Got questions? Get instant answers now!

Practice set b

Fill in the missing term.

- a + 2 - a + 3 = a - 3

a + 2

Got questions? Get instant answers now!

Exercises

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

- 11 x x + 1

x 1

Got questions? Get instant answers now!

x - 1 x 2 - 4

x 2 , 2

Got questions? Get instant answers now!

- x + 4 x 2 - 36

x 6 , 6

Got questions? Get instant answers now!

2 b b ( b + 6 )

b 0 , 6

Got questions? Get instant answers now!

3 x + 4 x x - 10 x + 1

x 0 , 10 , 1

Got questions? Get instant answers now!

6 a a 3 a - 5 7 - a

x 0 , 5 , 7

Got questions? Get instant answers now!

- 8 b 2 - 4 b + 3

b 1 , 3

Got questions? Get instant answers now!

y - 9 y 2 - y - 20

y 5 , 4

Got questions? Get instant answers now!

2 x + 7 6 x 3 + x 2 - 2 x

x 0 , 1 2 , 2 3

Got questions? Get instant answers now!

- x + 4 x 3 - 8 x 2 + 12 x

Got questions? Get instant answers now!

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

- 3 5 and - 3 5

( 3 ) 5 = 15 , ( 3 · 5 ) = 15

Got questions? Get instant answers now!

1 4 and 1 4

( 1 · 4 ) = 4 , 4 ( 1 ) = 4

Got questions? Get instant answers now!

- 9 10 and 9 - 10

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

Got questions? Get instant answers now!

For the following problems, fill in the missing term.

- 3 x + 4 2 x - 1 = 2 x - 1

Got questions? Get instant answers now!

- 2 x + 7 5 x - 1 = 5 x - 1

2 x 7

Got questions? Get instant answers now!

- x - 2 6 x - 1 = 6 x - 1

Got questions? Get instant answers now!

- x - 4 2 x - 3 = 2 x - 3

x + 4

Got questions? Get instant answers now!

- a + 1 - a - 6 = a + 6

a + 1

Got questions? Get instant answers now!

y + 10 - y - 6 = y + 6

y 10

Got questions? Get instant answers now!

Exercises for review

( [link] ) Write ( 15 x - 3 y 4 5 x 2 y - 7 ) - 2 so that only positive exponents appear.

Got questions? Get instant answers now!

( [link] ) Solve the compound inequality 1 6 x - 5 < 13 .

1 x < 3

Got questions? Get instant answers now!

( [link] ) Factor 8 x 2 - 18 x - 5 .

Got questions? Get instant answers now!

( [link] ) Factor x 2 - 12 x + 36 .

( x 6 ) 2

Got questions? Get instant answers now!

( [link] ) Supply the missing word. The phrase "graphing an equation" is interpreted as meaning "geometrically locate the to an equation."

Got questions? Get instant answers now!

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask