EQN-03-10

Simplifying Algebraic Fractions

Cancel common factors safely and combine algebraic fractions over a common denominator.

~5 min read•Difficulty 3/5

You may cancel a factor common to the whole top and whole bottom — never part of a sum.

(6 x) / (3 x) = 2 ✅ (3 x is a factor of both)

(x + 2) /2 = x + 1 ❌ you cannot cancel the 2 in x + 2

Factor first, then cancel.

(x^{2} - 4) / (x - 2) = (x - 2) (x + 2) / (x - 2) = x + 2

Multiply straight across; to divide, multiply by the reciprocal — just like numeric fractions.

(a / b) \cdot (c / d) = ac / bd (a / b) \div (c / d) = (a / b) (d / c)

Use a common denominator first.

1/ x + 1/ y = y / (xy) + x / (xy) = (x + y) / (xy)

Most "simplify" questions hide a difference of squares or a common factor. Factor the numerator and denominator before doing anything else.

Worked examples

Example 1 (difficulty 2/5)

Simplify

: (x^{2} - 9) / (x + 3) .

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Simplify

: 1/2 + 1/ x .

◎ Predict the answer first:

Factor the numerator and denominator before cancelling anything.
You can cancel a common FACTOR, never a single term inside a sum.
Add/subtract algebraic fractions over a common denominator, exactly like numbers.

Cancelling a term from a sum

, e . g . (x + 2) /2 = x

Why: It looks like the 2 divides out.

Fix: Only a factor common to the entire top and bottom cancels.

Adding fractions without a common denominator

Why: Tops and bottoms get added directly.

Fix: Rewrite both over the same denominator first.