What if you had a rational expression like ? How could you simplify it? After completing this Concept, you'll be able to reduce rational expressions like this one to their simplest terms and find their excluded values.
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CK-12 Foundation: 1207S Rational Expressions
Watch this video for more examples of how to simplify rational expressions.
YourTeacher: Simplifying Rational Expressions
Guidance
A simplified rational expression is one where the numerator and denominator have no common factors. In order to simplify an expression to lowest terms , we factor the numerator and denominator as much as we can and cancel common factors from the numerator and the denominator.
Simplify Rational Expressions
Example A
Reduce each rational expression to simplest terms.
a)
b)
c)
Solution
a)
b)
c)
When reducing fractions, you are only allowed to cancel common factors from the denominator but NOT common terms. For example, in the expression , we can cross out the factor because . But in the expression we can’t just cross out the terms.
Why can’t we do that? When we cross out terms that are part of a sum or a difference, we’re violating the order of operations (PEMDAS). Remember, the fraction bar means division. When we perform the operation , we’re really performing the division — and the order of operations says that we must perform the operations inside the parentheses before we can perform the division.
Using numbers instead of variables makes it more obvious that canceling individual terms doesn’t work. You can see that — but if we canceled out the 9’s first, we’d get , or -0.2, instead.
Find Excluded Values of Rational Expressions
Whenever there’s a variable expression in the denominator of a fraction, we must remember that the denominator could be zero when the independent variable takes on certain values. Those values, corresponding to the vertical asymptotes of the function, are called excluded values. To find the excluded values, we simply set the denominator equal to zero and solve the resulting equation.
Example B
Find the excluded values of the following expressions.
a)
b)
Solution
a)
b)
Removable Zeros
Removable zeros are those zeros from the original expression, but is not a zero for the simplified version of the expression. However, we have to keep track of them, because they were zeros in the original expression. This is illustrated in the following examples.
Example C
Determine the removable values of .
Solution:
Notice that in the expressions in Example A, we removed a division by zero when we simplified the problem. For instance, we rewrote as . The denominator of this expression is zero when or when .
However, when we cancel common factors, we simplify the expression to . This reduced form allows the value , so is its only excluded value.
Technically the original expression and the simplified expression are not the same. When we reduce a radical expression to its simplest form, we should specify the removed excluded value. In other words, we should write our final answer as .
Example D
Determine the removable values of the expressions from Example A parts b and c.
Solution:
We should write the answer from Example A, part b as .
The answer from Example A, part c as .
Watch this video for help with the Examples above.
CK-12 Foundation: Rational Expressions
Vocabulary
- Whenever there’s a variable expression in the denominator of a fraction, we must remember that the denominator could be zero when the independent variable takes on certain values. Those values, corresponding to the vertical asymptotes of the function, are called excluded values.
- Removable zeros are those zeros from the original expression, but is not a zero for the simplified version of the expression.
Guided Practice
Find the excluded values of .
Solution
Explore More
Reduce each fraction to lowest terms.
Find the excluded values for each rational expression.
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