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Operations with Rational Expressions

by Kourtney Kelley

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OPERATIONS
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WITH
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RATIONAL EXPRESSIONS
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KOURTNEY KELLEY
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What is a rational expression?
A rational expression is a quotient of two polynomials.

We must remember that a polynomial is the sum or difference of more than one term.

A rational expression may include algebraic expressions with variables in the numerator, denominator, or both.

Similar to rational numbers, rational expressions are closed under the four basic operations: addition, subtraction, mulitplication, and division.
Addition and Subtraction with Rational Expressions
3. Add or Subtract the expressions
Each expression must have the same denominator. Once this is completed, you may now add or subtract the numerators ONLY. The denominators stay they same.
1. Find a common denominator, called the LCD (Lease Common Denominator)
If your rational expressions are not the same, you must find a common denominator. This is called the Least Common Denominator. The LCD is the smallest (in value) expression that each denomintor in the addition or subtraction equation share.
2. Rewrite each expression with the LCD
After you have found the LCD, now rewrite each expression with the LCD. This may require you to multiple each expression within the equation so when you are finished they share the same denominator.
In this example, the LCD is 6x^2. Therefore, in order for each expression to have a denominator of 6x^2, each expression must be manipulated. After each expression has equivalent denominators, we now can add ONLY the numerators. Remeber, the denominators stay the same.
Addition with Rational Expressions
EXAMPLE
Notice the steps that the example to the left follows. First it simplfies each denominator by factoring. This helps in finding the LCD. Once the LCD is found, it then multiplies each expression to include the LCD. Finally it adds all numerators and the denomitaor stays the same.
Subtraction with Rational Expressions
EXAMPLE
Notice the steps that the example to the left follows. First it simplfies each denominator by factoring. This helps in finding the LCD. Once the LCD is found, it then multiplies only one the expressions to include the LCD, because the other already is the LCD. Finally it subtracts all numerators and the denomitaor stays the same.
Videos to assist with Addition and Subtraction of Rational Expressions
Subtraction with Rational Expressions
In this video below, taught by Khan Academy, they review several examples of how to subtract with Rational Expressions.
Addition with Rational Expressions
In this video above, taught by Khan Academy, they review several examples of how to add with Rational Expressions.
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