Write 3/(x − 2) − 2/(x + 1) as a single fraction in its simplest form.
- The common denominator is (x − 2)(x + 1).
- Top: 3(x + 1) − 2(x − 2) = 3x + 3 − 2x + 4
- 3x + 3 − 2x + 4 = x + 7
Answer: (x + 7)/((x − 2)(x + 1))
Algebraic fractions
Maths · Algebra and graphs. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
An algebraic fraction has letters on the top, the bottom or both. It obeys exactly the same rules as a fraction made of numbers.
To simplify, factorise the top and the bottom completely, then cancel any factor that appears in both. You may cancel only factors, which are things multiplied together. You may never cancel single terms that are added or subtracted.
To add or subtract, first write both fractions with a common denominator, then combine the tops over that one denominator. To multiply, multiply the tops and multiply the bottoms. To divide, turn the second fraction upside down and multiply. Cancel common factors whenever you can.
Rules to remember
- Simplify: factorise top and bottom, then cancel common factors, for example (x2 − 4)/(x2 + 2x) = (x − 2)(x + 2)/(x(x + 2)) = (x − 2)/x
- a/b + c/d = (ad + bc)/(bd)
- a/b − c/d = (ad − bc)/(bd)
- (a/b) × (c/d) = (ac)/(bd)
- (a/b) ÷ (c/d) = (a/b) × (d/c) = (ad)/(bc)
- When you subtract, keep the second top in brackets so that every sign changes
Common mistake
Students cancel terms instead of factors, for example cancelling the x2 in (x2 − 9)/(x2 + 3x). Factorise first and cancel only a whole bracket or a factor that multiplies everything.
Worked example
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