Algebraic fractions and proof (Higher tier)

Simplifying algebraic fractions

Factorise numerator and denominator, then cancel common factors.

Example: (x² − 9) / (x² − x − 6) = (x − 3)(x + 3) / [(x − 3)(x + 2)] = (x + 3) / (x + 2).

Adding/subtracting algebraic fractions

Find a common denominator (often the product, or LCM of the factors). Then combine numerators.

Example: 2/(x + 1) + 3/(x − 2) = [2(x − 2) + 3(x + 1)] / [(x + 1)(x − 2)] = (2x − 4 + 3x + 3) / [(x + 1)(x − 2)] = (5x − 1) / [(x + 1)(x − 2)].

Solving equations involving algebraic fractions

Multiply both sides by the common denominator to clear fractions, then solve as usual. Always check for invalid solutions (where a denominator would be zero).

Algebraic proof

Show identities or properties by algebraic manipulation.

Proving "an even number" or "a multiple of n"

Express in the form 2k (even), 2k + 1 (odd), n × k (multiple of n) for integer k.

Example: prove the sum of three consecutive integers is a multiple of 3. Let them be n, n + 1, n + 2. Sum = 3n + 3 = 3(n + 1). Since (n + 1) is an integer, 3(n + 1) is a multiple of 3. ∎

Proving an identity

Use ≡ (identity, true for all values). Manipulate both sides to the same form.

Example: prove (n + 1)² − n² ≡ 2n + 1. LHS = (n² + 2n + 1) − n² = 2n + 1 = RHS. ∎

Counterexamples

To disprove a statement, give a single counterexample.

Example: "n² + 1 is always prime" — counterexample n = 4: 17 is prime; n = 8: 65 = 5 × 13, not prime. Statement is false.

Common CCEA exam tip

For proofs:

• State what you are doing ("Let n be any integer").
• End with a clear conclusion line ("Therefore the sum is always divisible by 3").
• The B1/A1 final mark depends on the explicit conclusion.