Simplifying Algebraic Expressions Worked ExamplesIGCSE Maths 0580

The June 2025 examiner report noted that "errors were made in dealing with the negative term when simplifying" — here $-4q + q$ is $-3q$, not $-5q$. $p$ and $q$ terms are not like terms, so $-3p - 3q$ is the complete answer; do not spoil it by writing $-6pq$.

A common error is the second sign: writing $-2x - 8$ gives $4x - 23$. Multiply the sign in front of the bracket through every term inside it.

Check the constant term first as a quick sanity test: $-3 \times 5 = -15$ must appear. If your constant is wrong, the expansion is wrong.

A common error is forgetting to square the $3$, giving $6x^5y^7$. When a bracket is raised to a power, the coefficient is raised too.

$y$ on the bottom counts as $y^1$, so the index becomes $3 - 1 = 2$, not $3$. The June 2025 examiner report noted that, when multiplying terms, "dealing with the y terms sometimes resulted in an error" — the letter with the unwritten index $1$ is the one to watch.

Two common errors combine here: squaring term by term, and subtracting only the first term of the second bracket. Writing the bracket in at step 3 prevents both. Check at $x = 1$: $(5)^2 - (0)^2 = 25$, and $3 + 14 + 8 = 25$. ✓

The sign error in $q$ is the one to avoid: $(2x + 5) - (x - 1)$ is $x + 6$, not $x + 4$. A quick check for (b): every L-shape made only of right angles has the same perimeter as the rectangle around it, $2\big((3x + 2) + (2x + 5)\big) = 10x + 14$. ✓

Writing $(x - 1)^2$ as $x^2 + 1$ or $x^2 - 1$ loses the method mark in (a). The March 2026 examiner report found that quite a few candidates turned a squared bracket into the product of two different brackets, one with $+$ and one with $-$; $(x - 1)^2$ means $(x - 1)(x - 1)$. Check with $x = 4$: the rectangle is $11 \times 9 = 99$, the square is $3^2 = 9$, the shaded area is $90$, and $16 + 60 + 14 = 90$. ✓ In (b), adding the two new edges on top of the full rectangle perimeter is a common mistake.

Using the full difference $x + 2$ as the height doubles the area — the frame has a border on both sides of the hole. Squaring term by term, as in $(2x + 3)^2 = 4x^2 + 9$, loses the method mark in (a). Check with $x = 2$: outer side $7$, hole $3$, so $T = \frac{49 - 9}{4} = 10$, and $\frac{12 + 20 + 8}{4} = 10$. ✓

The March 2026 examiner report found that quite a few candidates changed a squared bracket such as $(x - 3)^2$ into $(x - 3)(x + 3)$ rather than $(x - 3)(x - 3)$. The June 2025 examiner report also noted that some candidates "incorrectly attempted to expand the three brackets by multiplying them out in a pairwise fashion": expand two brackets fully, then multiply the result by the third. Check at $x = 1$: $3 \times 9 = 27$ and $2 - 15 + 24 + 16 = 27$. ✓

The June 2025 examiner report noted that some candidates treated the terms in front of the bracket as if they were in a bracket themselves and expanded them — only $-2y$ multiplies the bracket. It also recorded candidates omitting to multiply the second term inside the bracket; here that loses the $+2y^4$. Indices add when you multiply: $y \times y^3 = y^4$.

The June 2025 examiner report found this type more challenging, noting that "some applied the power to just the algebraic part of the term", which leaves $27x^4y^6$. Another common error is multiplying $27$ by $\frac{2}{3}$ to get $18$. Work out the number with roots and powers: denominator $3$ means cube root, numerator $2$ means square.

The March 2026 examiner report recorded a common incorrect answer formed by adding a squared term and a linear term together, as if $2a^2 + 3a$ were $5a^2$. Here that would give $5m^2$. $m^2$ and $m$ are unlike terms, so the answer keeps both.