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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.
The following examples illustrate the use of the zero-factor property.
What value will produce zero in the expression ? By the zero-factor property, if , then .
What value will produce zero in the expression
? By the zero-factor property, if
, then
Thus,
when
.
What value(s) will produce zero in the expression
? By the zero-factor property, if
, then
Thus,
when
or
.
What value(s) will produce zero in the expression
? We must factor
to put it into the zero-factor property form.
Now,
when
Thus,
when
or
.
What value(s) will produce zero in the expression
? We must factor
to put it into the zero-factor property form.
Now,
when
Thus,
when
or
.
Find the domain of the following expressions.
.
The domain is the collection of all real numbers except 1. One is not included, for if
, division by zero results.
.
If we set
equal to zero, we find that
.
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.
.
Setting
, we find that
and
. 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.
.
Setting
, we get
Thus,
and
produce division by zero and must be excluded from the domain. The domain is the collection of all real numbers except –3 and 5.
.
Setting
, we get
. These numbers must be excluded from the domain. The domain is the collection of all real numbers except 0, 1, 3, –10.
.
Setting
, we get
=
,
. The domain is the collection of all real numbers except
and
.
.
No value of
is excluded since for any choice of
, the denominator is never zero. The domain is the collection of all real numbers.
.
No value of
is excluded since for any choice of
, the denominator is never zero. The domain is the collection of all real numbers.
Find the domain of each of the following rational expressions.
From our experience with arithmetic we may recall the equality property of fractions. Let , , , be real numbers such that and .
Two fractions are equal when their cross-products are equal.
We see this property in the following examples:
, since .
, since and .
Since , .
A useful property of fractions is the negative property of fractions .
To see this, consider
. Is this correct?
By the equality property of fractions,
and
. Thus,
. 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.
is best written as .
is best written as .
could be written as , which would then yield .
This expression seems less cumbersome than does the original (fewer minus signs).
Fill in the missing term.
For the following problems, find the domain of each of the rational expressions.
For the following problems, show that the fractions are equivalent.
For the following problems, fill in the missing term.
( [link] ) Write so that only positive exponents appear.
( [link] ) Factor .
(
[link] ) Supply the missing word. The phrase "graphing an equation" is interpreted as meaning "geometrically locate the
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