Algebraic Fractions & Proof

GCSE Maths · Algebra

Algebraic Fractions & Proof (Higher)

Simplifying Algebraic Fractions

Algebraic fractions follow the same rules as numerical fractions. To simplify, factorise the numerator and denominator, then cancel common factors.

Worked Example: Simplify (x² − 9) / (x² + 5x + 6).

Numerator: x² − 9 = (x + 3)(x − 3) (difference of two squares)
Denominator: x² + 5x + 6 = (x + 2)(x + 3)
Cancel (x + 3): (x − 3) / (x + 2)

Worked Example: Simplify (2x² + 6x) / (x² + 3x).

Numerator: 2x(x + 3)
Denominator: x(x + 3)
Cancel x(x + 3): 2

Adding and Subtracting Algebraic Fractions

Find a common denominator, then combine.

Worked Example: Write as a single fraction: 3/(x+1) + 2/(x−4).

Common denominator: (x + 1)(x − 4)
= 3(x − 4)/((x+1)(x−4)) + 2(x + 1)/((x+1)(x−4))
= (3x − 12 + 2x + 2) / ((x+1)(x−4))
= (5x − 10) / ((x+1)(x−4))
= 5(x − 2) / ((x+1)(x−4))

Multiplying and Dividing

Multiplying: Factorise first, cancel, then multiply across.

Dividing: Flip the second fraction and multiply.

Worked Example: Simplify (x² − 4)/(x + 1) ÷ (x + 2)/(x² + x).

Flip: (x² − 4)/(x + 1) × (x² + x)/(x + 2)
Factorise: ((x+2)(x−2))/(x+1) × (x(x+1))/(x+2)
Cancel (x+2) and (x+1): x(x − 2) or equivalently x² − 2x

Solving Equations with Algebraic Fractions

Multiply every term by the common denominator to clear the fractions.

Worked Example: Solve 5/(x−2) − 3/x = 1.

Multiply through by x(x − 2): 5x − 3(x − 2) = x(x − 2)
5x − 3x + 6 = x² − 2x
2x + 6 = x² − 2x
0 = x² − 4x − 6
Quadratic formula: x = (4 ± √(16 + 24))/2 = (4 ± √40)/2 = 2 ± √10

Algebraic Proof

An algebraic proof demonstrates that a statement is always true by using algebra.

Key representations:

Even number: 2n
Odd number: 2n + 1 (or 2n − 1)
Consecutive integers: n, n + 1, n + 2
Consecutive even numbers: 2n, 2n + 2, 2n + 4
Multiple of 3: 3n

Worked Example: Prove that the sum of any three consecutive odd numbers is always a multiple of 3.

Let the three consecutive odd numbers be (2n + 1), (2n + 3), (2n + 5)
Sum = (2n + 1) + (2n + 3) + (2n + 5) = 6n + 9 = 3(2n + 3)
Since 3(2n + 3) has a factor of 3, it is always a multiple of 3. ∎

Worked Example: Prove that (n + 3)² − (n − 2)² ≡ 5(2n + 1).

LHS = (n² + 6n + 9) − (n² − 4n + 4)
= n² + 6n + 9 − n² + 4n − 4
= 10n + 5
= 5(2n + 1) = RHS ∎

Exam Tips

Always factorise before cancelling — never cancel individual terms (x² + 3x)/(x) ≠ x + 3x — wait, actually that one does simplify to x + 3; the error is cancelling terms across a sum like (x + 3)/x → 1 + 3 = 4 — this is WRONG
In proofs, define your variables clearly at the start ("Let n be any integer")
End proofs with a concluding statement that links to the question
Do not skip the factorising step — it earns marks even if you get the final answer

Algebraic Fractions & Proof