What Are Algebraic Fractions?

An algebraic fraction is any fraction where the numerator or denominator contains a variable. Examples include x/3, (2x + 1)/(x - 4), and (x² - 9)/(x + 3). They follow the same arithmetic rules as plain number fractions, but you need to factor before simplifying.

One thing to keep in mind: the denominator can never equal zero. When you work with algebraic fractions, note which values of the variable would make the denominator zero and exclude them.

Simplifying Algebraic Fractions

Factor both the numerator and denominator completely, then cancel common factors. Do not cancel individual terms. You can only cancel things that are being multiplied across the full numerator and full denominator.

Simplify: (x² - 9) / (x + 3)

Factor the numerator: (x + 3)(x - 3) / (x + 3)

Cancel (x + 3):
= (x - 3), where x is not equal to -3

A common mistake is trying to cancel terms that are added or subtracted. For example, (x + 3)/(x + 5) cannot be simplified because x is a term, not a factor.

Adding and Subtracting Algebraic Fractions

Same as regular fractions: you need a common denominator first. If the denominators match, add or subtract the numerators directly. If not, find the lowest common denominator (LCD), convert each fraction, then combine.

Add: 3/x + 2/(x + 1)

LCD = x(x + 1)

3(x + 1) / x(x + 1) + 2x / x(x + 1)

= (3x + 3 + 2x) / x(x + 1)

= (5x + 3) / x(x + 1)

When subtracting, watch the signs. Distribute the minus sign across the entire numerator of the fraction being subtracted before combining like terms.

Multiplying Algebraic Fractions

Multiply numerator by numerator and denominator by denominator. Before you multiply, factor everything and cancel common factors. This keeps things smaller and avoids messy numbers later.

Multiply: (x² - 4) / (x + 3) * (x + 3) / (x + 2)

Factor: (x + 2)(x - 2) / (x + 3) * (x + 3) / (x + 2)

Cancel (x + 3) and (x + 2):

= (x - 2)

Dividing Algebraic Fractions

Dividing by a fraction is the same as multiplying by its reciprocal. Flip the second fraction, then multiply. Factor and cancel before multiplying.

For example, (a/b) divided by (c/d) = (a/b) * (d/c) = ad / bc. The same applies when a, b, c, and d are polynomial expressions.

Common Mistakes

  • Canceling terms instead of factors. You can only cancel something that is multiplied across the entire numerator and denominator.
  • Forgetting to note which values make the denominator zero.
  • Not fully factoring before simplifying. Partial factoring causes you to miss cancellations.
  • Sign errors when subtracting, particularly when the second fraction has a negative in the denominator.