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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 .

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 .

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 .

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 .

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 .

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.

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.

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.

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.

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.

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 .

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.

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.

Practice set a

Find the domain of each of the following rational expressions.

2 x 7

7

5 x x ( x + 4 )

0 , 4

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

2 , 1

5 a + 2 a 2 + 6 a + 8

2 , 4

12 y 3 y 2 - 2 y - 8

- 4 3 , 2

2 m - 5 m 2 + 3

All real numbers comprise the domain.

k 2 - 4 5

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 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 .

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

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

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 .

- y 9 is best written as - y 9 .

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

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

- 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).

Practice set b

Fill in the missing term.

- 5 y - 2 = y - 2

5

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

a + 2

- 8 5 - y = y - 5

8

Exercises

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

6 x - 4

x 4

- 3 x - 8

- 11 x x + 1

x 1

x + 10 x + 4

x - 1 x 2 - 4

x 2 , 2

x + 7 x 2 - 9

- x + 4 x 2 - 36

x 6 , 6

- a + 5 a a - 5

2 b b ( b + 6 )

b 0 , 6

3 b + 1 b b - 4 b + 5

3 x + 4 x x - 10 x + 1

x 0 , 10 , 1

- 2 x x 2 4 - x

6 a a 3 a - 5 7 - a

x 0 , 5 , 7

- 5 a 2 + 6 a + 8

- 8 b 2 - 4 b + 3

b 1 , 3

x - 1 x 2 - 9 x + 2

y - 9 y 2 - y - 20

y 5 , 4

y - 6 2 y 2 - 3 y - 2

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

x 0 , 1 2 , 2 3

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

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

- 3 5 and - 3 5

( 3 ) 5 = 15 , ( 3 · 5 ) = 15

- 2 7 and - 2 7

1 4 and 1 4

( 1 · 4 ) = 4 , 4 ( 1 ) = 4

- 2 3 and - 2 3

- 9 10 and 9 - 10

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

For the following problems, fill in the missing term.

- 4 x - 1 = x - 1

- 2 x + 7 = x + 7

2

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

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

2 x 7

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

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

x + 4

- x + 5 - x - 3 = x + 3

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

a + 1

x - 7 - x + 2 = x - 2

y + 10 - y - 6 = y + 6

y 10

Exercises for review

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

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

1 x < 3

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

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

( x 6 ) 2

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

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Source:  OpenStax, Algebra ii for the community college. OpenStax CNX. Jul 03, 2014 Download for free at http://cnx.org/content/col11671/1.1
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