Simplifying and Manipulating Algebraic Expressions — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers tidying algebraic expressions into their shortest correct form. By the end of this guide you should be able to collect like terms, multiply and divide algebraic terms, and apply the index laws to expressions containing powers.

You should also be able to expand a single bracket, simplify the result, factorise by taking out a common factor, and recognise when an expression is already as simple as it can be.

The organising idea is that letters behave exactly like units of measurement. You can add 3 metres to 5 metres and get 8 metres, but 3 metres plus 5 kilograms cannot be combined into anything shorter — and 3x + 5y behaves in precisely the same way. Multiplication is different: 3 metres times 5 metres gives 15 square metres, a new unit, just as 3x × 5x gives 15x². That comparison explains the whole topic — why only like terms collect, why x and x² never combine, and why multiplying changes the power while adding does not.

Key terms and definitions

Term — a part of an expression separated from the rest by a plus or minus sign.

Coefficient — the number in front of a letter. In 7x the coefficient is 7.

Like terms — terms with exactly the same letters raised to exactly the same powers.

Expression — a collection of terms with no equals sign.

Expand — multiply out a bracket.

Factorise — the reverse of expanding: put an expression back into brackets.

Index (plural indices) — a power, such as the 3 in x³.

Simplify — write an expression in its shortest equivalent form.

Core concepts

Collecting like terms

Like terms have the same letters raised to the same powers. Only they can be combined, and combining them means adding or subtracting the coefficients while the letter part stays unchanged.

So 4x + 3x = 7x, and 9y − 2y = 7y.

Unlike terms stay separate. In 5a + 2b − 3a, the a terms combine to give 2a, but the b term cannot join them, so the answer is 2a + 2b.

Crucially, x and x² are not like terms, any more than metres and square metres are the same unit. The expression x² + x is already fully simplified.

Nor are xy and x like terms. But xy and yx are the same thing, since multiplication can be done in any order, so 3xy + 2yx = 5xy.

Each term carries the sign in front of it. In 7a − 4b − 2a + 6b, the a terms are +7a and −2a giving 5a, and the b terms are −4b and +6b giving 2b, so the answer is 5a + 2b.

Multiplying terms

Multiply the numbers together and the letters together.

For 3x × 4y: the numbers give 12 and the letters give xy, so the answer is 12xy.

When the same letter appears twice, the powers add: 2a × 5a = 10a², because a × a = a².

With several letters, deal with each separately: 4x²y × 3xy³ gives 12 for the numbers, x³ for the x parts, and y⁴ for the y parts, so the answer is 12x³y⁴.

Signs follow the usual rules: −3x × 4y = −12xy, and −2a × −5a = 10a².

Dividing terms

Divide the numbers and subtract the indices of matching letters.

For 12x⁵ ÷ 3x²: the numbers give 4 and the powers give x³, so the answer is 4x³.

Written as a fraction, this is cancelling: 12x⁵ over 3x² cancels to 4x³.

Where a letter disappears entirely, it has become 1 and simply vanishes from the answer: 6ab ÷ 2a = 3b.

The index laws

Three rules cover everything on this topic.

Multiplying: add the indices. x⁴ × x³ = x⁷.

Dividing: subtract the indices. x⁶ ÷ x² = x⁴.

A power of a power: multiply the indices. (x³)² = x⁶.

Two special cases follow from them. Anything to the power zero is 1, since x³ ÷ x³ must be both 1 and x⁰. And a negative index means a reciprocal, so x⁻² is 1 over x².

Take care that these apply only to matching letters. There is no way to simplify x³ × y², because the bases differ.

Coefficients are handled separately from indices: 3x² × 4x³ gives 12 from the numbers and x⁵ from the powers, so 12x⁵. Adding the coefficients as well as the indices is a common muddle.

When a power applies to a whole product, it reaches everything inside: (2x)³ = 2³ × x³ = 8x³, not 2x³.

Expanding a single bracket

Multiply every term inside the bracket by the term outside.

So 3(x + 4) = 3x + 12, and 5(2a − 3) = 10a − 15.

A negative outside the bracket changes every sign inside: −2(x − 5) = −2x + 10. The second sign change is the one most often missed.

With a letter outside, the index laws apply: x(x + 3) = x² + 3x.

Expanding and then simplifying

Many questions expand two brackets and then ask for the result in its simplest form, which means collecting like terms afterwards.

For 3(x + 2) + 4(x − 1): expanding gives 3x + 6 + 4x − 4, and collecting gives 7x + 2.

Where the second bracket is subtracted, the minus applies to everything inside it. For 5(x + 3) − 2(x + 1): expanding gives 5x + 15 − 2x − 2, so the answer is 3x + 13. Writing −2x + 2 there is the standard error.

Factorising with a common factor

Factorising reverses expanding: look for the highest factor common to every term and take it outside a bracket.

For 6x + 9, the highest common factor is 3, giving 3(2x + 3).

Letters count as factors too. For x² + 5x, every term contains x, so it factorises to x(x + 5).

Both together: 8a²b + 12ab² has a common factor of 4ab, giving 4ab(2a + 3b).

Always check by expanding your answer mentally — it should return the original expression — and make sure you took out the highest common factor, since 2(3x + 6) is factorised but not fully.

When an expression is already simplified

Some expressions cannot be shortened, and recognising that saves time.

x + y is finished, since the terms are unlike. So is 3a² + 2a, and so is 5x + 7.

If no two terms share exactly the same letters and powers, and there is no bracket to expand or common factor to remove, the expression is already in its simplest form.

Worked examples

Example 1: Collecting like terms

Simplify 8x + 3y − 5x + 2y − 4.

Group the terms by type, keeping the sign in front of each.

The x terms are +8x and −5x, giving 3x.

The y terms are +3y and +2y, giving 5y.

The number term is −4 and has nothing to pair with.

The answer is 3x + 5y − 4, which cannot be shortened further because the three terms are unlike.

Example 2: Expanding and simplifying with a subtraction

Expand and simplify 4(2x + 5) − 3(x − 2).

Expand the first bracket: 8x + 20.

Expand the second, remembering the minus applies to both terms inside: −3x + 6.

Write them together: 8x + 20 − 3x + 6.

Collect like terms: 5x + 26.

So the answer is 5x + 26. Treating the second bracket as −3x − 6 would have given 5x + 14.

Example 3: Index laws and factorising

Simplify 10a⁴b ÷ 2a²b, then factorise 6x² + 15x.

For the division: the numbers give 5, the a powers subtract to give a², and the b powers cancel completely to give 1. So the answer is 5a².

For the factorising: the highest common factor of 6 and 15 is 3, and both terms contain x, so the common factor is 3x.

Taking it out gives 3x(2x + 5).

Checking by expanding: 3x × 2x = 6x² and 3x × 5 = 15x. ✓

Common mistakes and how to avoid them

Combining unlike terms. x and x² never combine, nor do x and y. Think metres and kilograms.

Adding indices when adding terms. x² + x² = 2x², not x⁴. The indices only add when terms are multiplied.

Adding coefficients when multiplying. 3x² × 4x³ = 12x⁵ — multiply the numbers, add the powers.

Missing the second sign change. Subtracting a bracket changes every sign inside it.

Expanding a bracket incompletely. The term outside multiplies every term inside.

Taking out a factor that is not the highest. 2(3x + 6) is not fully factorised; 6(x + 2) is.

Applying a power to only part of a product. (2x)³ = 8x³, not 2x³.

Exam technique for "Simplifying Expressions"

Group like terms visually before combining them — underlining or circling each type prevents terms being missed or counted twice.

Keep the sign in front of each term as part of that term. Most collecting errors are sign errors.

Expand brackets fully as a separate written line before collecting, rather than doing both at once.

After factorising, expand your answer to check it returns the original expression. It takes seconds.

Check the common factor you removed is the highest one available, since partial factorising rarely earns full marks.

If two terms look combinable, test whether their letters and powers match exactly. If not, leave them alone — an answer that is already simplified is a valid answer.

Quick revision summary

Letters behave like units. Only like terms — same letters, same powers — can be collected, by adding or subtracting the coefficients. So 4x + 3x = 7x, but x² + x is already finished.

To multiply terms, multiply the numbers and add the indices of matching letters: 2a × 5a = 10a². To divide, divide the numbers and subtract the indices: 12x⁵ ÷ 3x² = 4x³.

Index laws: multiplying adds, dividing subtracts, and a power of a power multiplies. Anything to the power zero is 1, and a negative index means a reciprocal. A power outside a bracket reaches everything inside, so (2x)³ = 8x³.

To expand, multiply every term inside the bracket by the term outside. A subtracted bracket changes every sign inside it.

To factorise, take out the highest common factor, including letters: 6x² + 15x becomes 3x(2x + 5). Check by expanding.

An expression with no like terms, no brackets and no common factor is already simplified.