The Fractions Operation of Algebraic Expressions

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Operations with Algebraic Fractions

There are many techniques that will simplify your work as you perform operations with algebraic fractions. As you review these examples, note the steps involved in each operation and any methods that will save you time.

Reducing algebraic fractions

To reduce an algebraic fraction to lowest terms, first factor the numerator and the denominator; then cancel, (or divide out) common factors.
Example 1: Reduce.

• Warning: Do not cancel through an addition or subtraction sign as shown here.
  

Multiplying algebraic fractions

To multiply algebraic fractions, first factor the numerators and denominators that are polynomials; then, cancel where possible. Multiply the remaining numerators together and denominators together. (If you've canceled properly, your answer will be in reduced form.)
Example 2: Multiply.

Dividing algebraic fractions

To divide algebraic fractions, invert the fraction and multiply. Remember, you can cancel only after you invert.
Example 3: Divide.

Adding or subtracting algebraic fractions

To add or subtract algebraic fractions having a common denominator, simply keep the denominator and combine (add or subtract) the numerators. Reduce if possible.
Example 4: Perform the indicated operation.


To add or subtract algebraic fractions having different denominators, first find a lowest common denominator (LCD), change each fraction to an equivalent fraction with the common denominator, then combine each numerator. Reduce if possible. Example 5: Perform the indicated operation.


If there is a common variable factor with more than one exponent, use its greatest exponent. Example 6: Perform the indicated operation.


To find the lowest common denominator, it is often necessary to factor the denominators and proceed as follows. Example 7: Perform the indicated operation.

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