Factorisation Worked ExamplesIGCSE Maths 0580

Take out the highest common factor — number and letter. The June 2025 examiner report noted that most candidates were correct, "with just a few giving a partial factorisation". $3(2x^2 + 3x)$ and $x(6x + 9)$ are both partial and earn only one mark. Expand back to check.

This only works for a difference. $9x^2 + 25$ does not factorise. Spotting the pattern is worth practising — it appears in simplifying fractions and solving equations as well as here.

A negative constant always means one positive and one negative number. A positive constant with a negative middle term means both are negative. Reading the signs first narrows the search immediately.

If the two brackets at step 4 do not match, you split the middle term wrongly — go back rather than forcing it. The order of the split does not matter: $-4x + 15x$ gives the same answer.

The question just says 'Factorise', but full marks need the expression factorised completely. The March 2025 examiner report found that, after taking out the common factor, fewer candidates then recognised the bracket as a difference of two squares. Stopping at $2x(x^2 - 16)$ earns one mark out of two. Always check whether the bracket left behind factorises again.

Choose what to take out of the second pair so that the bracket matches the first. The June 2025 examiner report found "the most common error was to factorise the first two terms and then factorise the final two terms without repeating the same bracket". Another error it recorded was stopping at a form like $a(2x + 3y) - 2b(2x + 3y)$, which is not yet a product. Note the minus sign carried into the bracket at step 1; taking out $2b$ rather than $-2b$ depends on it. Check at $a=1, b=2, x=3, y=4$: original $= 6 + 12 - 24 - 48 = -54$, and $(1-4)(6+12) = -54$. ✓

The diagram is the grid method run backwards, and it matches splitting the middle term: $13x = 9x + 4x$, where $9 \times 4 = 36 = 6 \times 6$. Always expand your brackets to check. For example, $(6x + 3)(x + 2)$ gives $6x^2 + 15x + 6$, so it is wrong.

Taking the rectangle as $L \times r$ rather than $L \times 2r$ is the most common error in (a). $r(2L + \pi r)$ is fully factorised; taking out $\pi r$ would leave $\frac{2L}{\pi}$ in the bracket, which is not factorising. In (c), finding $L$ first ($105.75\ldots$ m) works but adds a step and a chance of rounding too early.

'Without expanding' means the mark is for the difference-of-two-squares step. Expanding both squares gets the same answer but does not earn it. A common slip is $(2x + 1) - (2x - 1) = 0$; subtracting $-1$ gives $+1$, so the bracket is $2$. In (c), squaring $10.1$ and $9.9$ by hand invites errors, but the factorised form is mental arithmetic.

The June 2025 examiner report noted that some candidates completed the first stage correctly but "could not deal with the factor of 1 for the second term". Writing the $1$ in, as in step 2, stops it disappearing: $(5a)(b - 2c)$ is wrong. The March 2025 examiner report also found that the most common incorrect answers came from sign errors in brackets like these. Expand to check: $(5a - 1)(b - 2c) = 5ab - 10ac - b + 2c$. ✓

The June 2025 examiner report noted: "Many did not realise they needed to start by equating the equation of the curve to 0." It also found a large number of candidates setting the derivative equal to zero instead; that finds the turning points ($x = \pm 2$ here), not where the curve crosses the axis. A decimal such as $3.46$ is not exact.

The June 2025 examiner report noted that many candidates did not use the factorised form from the previous part but used trial and error. 'Use your answer to part (a)' is the hint: a product of two whole numbers equal to $30$ means a factor pair. The limits $x + 2 \geqslant 3$ and $y + 4 \geqslant 5$ decide which pairs are allowed.