Mr Kidman's Classroom
Revise the key ideas
Higher — simplifying algebraic fractions
- Factorise numerator and denominator before cancelling common factors. = = x − 3, for x ≠ −3.
- Cancel factors, not individual terms in sums. is 1 + , not 3. The denominator cannot be zero.
- Record restrictions from the original denominator even when its factors disappear after cancellation. A simplified expression does not restore excluded values.
In the first example, x = −3 would make the original denominator x + 3 equal zero. The simplified expression x − 3 has a value there, but it cannot replace the original fraction at that excluded input.
- Worked example: = , with x ≠ 0. Divide coefficients by 3 and cancel a common factor x.
Higher — the four operations
- To multiply, factorise and cancel across the complete numerator and denominator: × = x + 2, with x ≠ −1 and x ≠ 2.
- To divide, multiply by the reciprocal of the second fraction. Check that the divisor itself is not zero as well as checking all denominators.
- Worked example: ÷ = × = , where x ≠ 0.
- For addition or subtraction, use a common denominator and combine whole numerators. + = = .
Here x cannot be 0 or −1: those values make one of the original denominators zero.
- Worked example: − = = , with x ≠ ±1. Bracket the subtracted numerator.
Higher — solving equations
- Multiply both sides of an equation by a common denominator to remove fractions, while preserving the restrictions on x.
- Worked example: = gives 2(x + 1) = 3x, so x = 2. This is allowed since x is neither 0 nor −1.
- Solutions obtained after clearing denominators must be checked in the original equation. Reject values that cause division by zero, even if they satisfy the rearranged polynomial.
- If a quadratic appears, solve it by factorisation or the quadratic formula and check that every candidate is an allowed value of x in the original equation.
Further reasoning
- A quadratic fraction — Solve 3/(x − 1) = x + 1. First exclude x = 1. Multiplying both sides by x − 1 gives 3 = (x + 1)(x − 1) = x² − 1, so x² = 4 and x = 2 or −2. Both are allowed: substitution gives 3 = 3 for x = 2 and −1 = −1 for x = −2. Clearing denominators can produce a quadratic; retain every candidate until it has been checked against the original restriction.
