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 |
Cite this article as:
McAdams, David E. Rational Expression. 5/5/2011. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/r/rationalexpression.html.
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Revision History
11/10/2009: Added addition, subtraction, multiplication and division of rational expressions. (
McAdams, David E.)
1/20/2009: Initial version. (
McAdams, David E.)