Manipulate algebraic expressions and algebraic fractions: simplify, add, subtract, multiply and divide them, solve equations involving them, and construct simple algebraic proofs.

What this dot point is asking

Algebraic manipulation is the foundation of the whole Pure unit, and CCEA GCSE Further Mathematics pushes it well beyond ordinary GCSE. You must be fluent with algebraic fractions: simplifying them by factorising and cancelling, combining them with the four operations, solving equations that contain them, and using algebra to prove general results. Every later Pure topic, from quadratics to calculus, leans on these skills, so they have to be automatic and accurate.

Simplifying algebraic fractions

An algebraic fraction simplifies only through common factors, so the first move is always to factorise the numerator and denominator completely. Look for a common factor, a difference of two squares $a^2 - b^2 = (a - b)(a + b)$, or a quadratic that factorises into two brackets. Once both parts are in factor form, cancel any factor that appears top and bottom.

For example, $\dfrac{6x^2 - 6}{3x + 3} = \dfrac{6(x^2 - 1)}{3(x + 1)} = \dfrac{6(x - 1)(x + 1)}{3(x + 1)} = 2(x - 1)$. The danger is cancelling parts of a sum: in $\dfrac{x + 2}{x + 4}$ nothing cancels, because $x$ is a term, not a factor.

The four operations on fractions

The rules mirror ordinary number fractions, but you must keep brackets intact.

• Multiply: multiply numerators together and denominators together, then cancel. Cancelling before multiplying keeps the numbers small.
• Divide: turn the second fraction upside down and multiply (multiply by the reciprocal).
• Add or subtract: rewrite both fractions over a common denominator, usually the product of the denominators or their lowest common multiple, then combine the numerators, expanding carefully.

Solving equations with algebraic fractions

When an equation contains fractions, clear them in one step by multiplying every term by the lowest common denominator. This turns the equation into a linear or quadratic one you already know how to solve.

Algebraic proof

Proof questions ask you to show a statement is always true, not just to check examples. The technique is to write the general object algebraically, manipulate it, and reach a form that makes the conclusion obvious. An even number is $2n$, an odd number is $2n + 1$, and consecutive integers are $n, n + 1, n + 2$, where $n$ is any integer.

For example, to prove the sum of three consecutive integers is a multiple of $3$, write them as $n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)$. Because the result is $3$ times an integer, it is always a multiple of $3$. The key is that a single algebraic argument covers every case at once, which is exactly the reasoning CCEA rewards.

Why this matters

These manipulation skills are the language of the entire Pure unit. Algebraic fractions reappear when you differentiate, integrate, work with functions, and rearrange formulae; difference-of-two-squares and quadratic factorising drive the polynomial and calculus topics; and the discipline of arguing generally underlies every proof you will meet. Marks here are method marks elsewhere, so building speed and accuracy now pays off across the whole paper.