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The definition of rational expressions and their domains are examined.
Definition
A rational expression is the quotient of two polynomials.
Examples of Rational Expressions
-
x - 1
-------
x + 2
-
x + 1
---------------
(x + 2)(x - 3)
-
1
--------
x2 + 1
-
x2 + 4
--------------
x3 + 2x2 - 3x
Domain of Rational Expressions
It is clear that the above expressions are undefined if a division by 0 occurs. The domain of a rational expression excludes all values that make the denominator equal to 0.
-
The domain of the rational expression
x - 1
-------
x + 2
is the set of all real numbers except x = -2
-
The domain of the rational expression
x + 1
---------------
(x + 2)(x - 3)
is the set of all real numbers except x = -2 and x = 3
-
The domain of the rational expression
1
--------
x2 + 1
is the set of all real numbers since the denominator x2 + 1 cannot be 0 for any real number x.
-
To find the domain of the rational expression,
x2 + 4
--------------
x3 + 2x2 - 3x
we first have to factor the denominator and find its zeros.
x3 + 2x2 - 3x = x(x2 + 2x - 3)
= x(x - 1)(x + 3)
The domain of the above rational expressions is the set of all real numbers except x = 0, x = 1 and x = - 3.
Exercises
Find the domain of each of the Rational Expressions given below.
-
x + 9
-------
x - 10
-
1
---------------
(x - 9)(x + 1)
-
1
--------
x3 + 1
-
-x + 7
--------
x4 - 16
Answers to the above exercises
More references to topics related to rational expressions.
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