Algebraic Expressions • Topic 4 of 4

Algebraic Fractions

An algebraic fraction is a ratio of expressions, and the same rules that govern number fractions apply. To simplify, factor the top and bottom, then cancel any factor they share — (x² − 9)/(x + 3) factors to (x + 3)(x − 3)/(x + 3) = x − 3. You can only cancel whole factors, never individual terms: the x in (x + 2)/x does not cancel. To multiply, factor everything and cancel across the fraction bar; to divide, flip the second fraction and multiply. To add or subtract, put both parts over a common denominator first. One caution the GRE quietly exploits: a value that makes any original denominator zero is excluded, so a cancelled expression is only equal to the original where the denominator was already nonzero.

✅ Solved examples

1. Simplify (x² − 25)/(x − 5).
Factor the top: (x + 5)(x − 5)/(x − 5). Cancel (x − 5): = x + 5 (for x ≠ 5).
2. Simplify (6x² + 3x)/(2x + 1).
Factor the top: 3x(2x + 1)/(2x + 1). Cancel (2x + 1): = 3x.
3. Multiply (x/3) · (9/x²).
= 9x / 3x² = 3/x (cancel a common 3 and one x).
4. Add (2/x) + (3/(2x)).
Common denominator 2x: 4/(2x) + 3/(2x) = 7/(2x).

✏️ Practice — try these, take hints as needed

1. Simplify (x² − 16)/(x + 4).
Factor the numerator.
(x + 4)(x − 4).
Cancel (x + 4).
x − 4
2. Simplify (4x + 8)/(x + 2).
Factor out 4 on top.
4(x + 2).
Cancel (x + 2).
4
3. Multiply (x²/4) · (2/x).
Multiply tops and bottoms.
2x²/4x.
Cancel.
x/2
4. Divide (x/6) ÷ (x/3).
Flip the second fraction.
(x/6)·(3/x).
Cancel x, then 3/6.
1/2
5. Subtract (5/(2x)) − (1/x).
Common denominator 2x.
1/x = 2/(2x).
5/(2x) − 2/(2x).
3/(2x)

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Common Factoring Patterns Take the chapter test →