Rational Expression
Pronunciation: /ˈræʃ.nl ɪkˈsprɛ.ʃən/ Explain
Simplifying Rational Expressions
Simplifying a rational expression is similar to
reducing a fraction.
Both simplifying a rational expression and reducing fractions involve finding
common factors
and canceling
those factors.
Example 1
Step | Equation | Description |
1 | |
This is the rational expression to simplify. |
2 | |
Factor both of the polynomials. |
3 | |
Cancel common factors and place any restrictions on the resulting
expression. |
Example 1 |
Example 2
Step | Equation | Description |
1 | |
This is the rational expression to simplify. |
2 | |
Transform the 2 to a fraction in order to find
the common denominator. |
3 | |
Multiply the 2 by
( x - 2 ) / ( x - 2 ) so the
fractions can be combined. |
4 | |
Multiply the first two fractions. |
5 | |
As the fractions now have a common denominator, combine the fractions. |
6 | |
Use the commutative law of addition to put like terms next to each other. |
7 | |
Combine like terms. |
8 | |
Factor the numerator and the denominator. |
9 | |
Cancel common factors and place any restrictions on the resulting expression. |
Example 2 |
Addition and Subtraction of Rational Expressions
Two or more rational expressions can added or subtracted.
Addition and subtraction of rational expressions is
similar to addition and subtraction of fractions:
The steps are:
- Find the least common denominator.
- Transform both rational expressions to the same
common denominator.
- Add or subtract the rational expressions by adding or subtracting the
numerators
and using the least common denominator.
Example 3
Step | Equation | Description |
1 | | This is the original expression. |
2 | The prime factorization
of x - 1 is
x - 1. |
To find the common denominator, find the prime factorization of the
denominator of the first expression. |
3 |
The prime factorization of x + 1 is
x + 1. |
Find the prime factorization of the denominator of the second
expression. |
4 |
The factor x + 1 is not found in the
second denominator. The second rational expression will be multiplied by
. |
Identify the factors of the second denominator not in the first denominator.
The first rational expression will be multiplied by a ratio of these factors. |
5 | |
Multiply the first rational expression by the factors identified in step
4. |
6 |
The factor x - 1 is not found in the
first denominator. The first rational expression will be multiplied by
. |
Identify the factors of the first denominator not in the second denominator.
The second rational expression will be multiplied by a ratio of these
factors. |
7 | |
Multiply the second rational expression by the factors identified in step
6. |
8 | |
Transform the expressions using the multiplied rational expressions. |
9 | |
Add the numerators, and copy the denominator. |
10 | |
Simplify the numerator by combining like terms. |
11 | |
This is the answer. |
Example 3 - Addition of rational expressions |
Example 4
Step | Equation | Description |
1 | |
This is the original expression. |
2 |
The prime factorization of x is
x. |
To find the common denominator, find the prime factorization of the
denominator of the first expression. |
3 |
The prime factorization of x^{2} +
x is x( x + 1
). |
Find the prime factorization of the denominator of the second
expression. |
4 | The factor x is found
in the second denominator. The second rational expression is already common. |
Identify the factors of the second denominator not in the first denominator.
The first rational expression will be multiplied by a ratio of these
factors. |
5 | The factor x is found in
the first denominator. The factor x + 1
is not found in the first denominator. The first rational expression
will be multiplied by . |
Identify the factors of the first denominator not in the second denominator.
The second rational expression will be multiplied by a ratio of these factors. |
6 | |
Multiply the second rational expression by the factors identified in step 6. |
7 | |
Transform the expressions using the multiplied rational expressions. |
8 | |
Add the numerators, and copy the denominator. |
9 | |
Simplify the numerator by combining like terms. |
10 | |
This is the answer. |
Example 4 - Subtraction of rational expressions |
Multiplication and Division of Rational Expressions
Two or more rational expressions can multiplied. Multiplication
and division of rational expressions is similar to
multiplication and division of fractions:
The steps are:
- Factor the rational expressions.
- Cancel common factors.
- Multiply the numerators.
- Multiply the denominators.
Example 5
Step | Equation | Description |
1 | | This is the expression to
multiply |
2 | | Factor the numerators. This will
allow you to more easily cancel common factors. |
3 | | Factor the denominators. This will
allow you to more easily cancel common factors. |
4 | | Cancel common factors. |
5 | | Multiply the numerators and multiply
the denominators. |
6 | | This is the answer. |
Example 5 - Multiplication of rational
expressions |
To divide rational expressions, multiply the divided by the reciprocal of the divisor.
Example 6
Step | Equation | Description |
1 | | This is the expression to divide. |
2 | | Multiply by the reciprocal of the
divisor. |
3 | | Factor the numerators. This will
allow you to more easily cancel common factors. |
4 | | Factor the denominators. This will
allow you to more easily cancel common factors. |
5 | | Cancel common factors. |
6 | | Multiply the numerators and multiply
the denominators. |
7 | | This is the answer. |
Example 6 - Division of rational expressions |
References
- McAdams, David E.. All Math Words Dictionary, rational expression. 2nd Classroom edition 20150108-4799968. pg 150. Life is a Story Problem LLC. January 8, 2015. Buy the book
Cite this article as:
McAdams, David E. Rational Expression. 4/30/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/r/rationalexpression.html.
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Revision History
4/30/2019: Changed equations and expressions to new format. (
McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
12/4/2018: Removed broken links, updated license, implemented new markup. (
McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (
McAdams, David E.)
11/10/2009: Added addition, subtraction, multiplication and division of rational expressions. (
McAdams, David E.)
1/20/2009: Initial version. (
McAdams, David E.)