What Counts as an Algebraic Fraction?

Any fraction with a variable somewhere in it, top, bottom, or both, qualifies as an algebraic fraction - x/3, (2x + 1)/(x - 4), and (x² - 9)/(x + 3) are all examples. The arithmetic doesn't change from what you'd do with plain numbers; the only new wrinkle is factoring before you simplify.

One rule never bends: the denominator can't be zero. So before simplifying or solving with an algebraic fraction, it's worth pausing to figure out which variable values would zero out the denominator, since those get excluded from the final answer.

Simplifying an Algebraic Fraction

Factor the numerator and denominator completely, then cancel any factors that appear in both. Notice the word "factors" - you can only cancel something multiplied across the whole numerator and the whole denominator, not a single term buried inside a sum.

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 error is canceling terms that are added or subtracted instead of multiplied. (x + 3)/(x + 5), for instance, doesn't simplify at all - those x's are terms, not shared factors, so nothing cancels.

Adding and Subtracting Algebraic Fractions

As with numeric fractions, you need matching denominators before you can add or subtract. When they already match, just combine the numerators. When they don't, find the lowest common denominator (LCD), rewrite each fraction over it, 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)

Subtraction is where sign mistakes tend to sneak in. Make sure the negative sign gets distributed across every term of the numerator you're subtracting before you start combining anything.

Multiplying Algebraic Fractions

Multiplying is the simplest operation here: numerator times numerator, denominator times denominator. Factor everything first and cancel any shared factors before multiplying - it keeps the numbers smaller and saves you from expanding something you'd only have to factor again.

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 as usual, factoring and canceling wherever possible before you expand anything.

In general, (a/b) divided by (c/d) equals (a/b) times (d/c), giving ad / bc. That holds whether a, b, c, and d are plain numbers or full polynomial expressions.

Mistakes Worth Watching For

  • Canceling a term instead of a factor - only something multiplied across the entire numerator and denominator is fair game to cancel.
  • Forgetting to note which variable values would make the denominator zero.
  • Stopping partway through factoring, which hides cancellations that were right there.
  • Sign slips during subtraction, especially when the fraction being subtracted already has a negative in its own denominator.