Defining algebraic fractions

Algebraic fractions are such that in . Remember that "a" is called the "numerator", while "b" is called the "denominator". Division by zero is undefined. In other words, division by 0 is meaningless (try dividing a cookie among 0 people in your head), therefore the denominator should never equal 0.


It's not always necessary to state the condition that "b ≠ 0" every step of the way. However, you may want to state it clearly as part of your answer if your final answer is an algebra fraction.

Keeping the fact that "b ≠ 0" in mind can help you solve some classes of algebraic problems, e.g. in determining asymptotes.

Addition & Subtraction of algebra fractions

To add or subtract algebra fractions having the same value denominator, just add or subtract the numerators as it is. Simplify the fractions in your final answer, if necessary.


To add or subtract algebra fractions having different denominators, first find the lowest common denominator, then change each fraction to use the lowest common denominator accordingly. Finally add or subtract each numerator. Reduce the fractions if necessary.


More complex examples

For (x + 1)(x + 3), you can use the following:
x² – x – 6 = (x – 3)(x + 3)

Multiplication of algebra fractions

To multiply algebra fractions, you'll likely need to first factorize the numerators and denominators that are polynomials. Simplify each term that are fractions where possible (this should make it easier to work with in the next step). Then do the multiplication process. Simplify the final fraction if necessary.


Division of algebra fractions

To divide algebra fractions, invert the second fraction (the numerator becomes the denominator and vice versa). Simplify the fractions if it will make them easier to work with. Then multiply both fractions. Yes, I really meant multiply them. As usual, be sure to simplify your answer if possible.


Algebra Fractions: Exercise


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