Checkpoint 8 Maths · Algebra · Expressions

Algebraic Expressions: Simplifying, Expanding and Factorising

What letters mean in expressions, formulae and equations; simplifying with the laws of arithmetic (including squares and cubes); expanding a single bracket and factorising by taking out the highest common factor.

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In the curriculum

Understand that a letter can mean different things: in an expression it can stand for any number, in a formula it stands for a quantity linked to other quantities, and in an equation it stands for an unknown number you can find (8Ae.01)
Use the laws of arithmetic and the order of operations with algebra — adding, subtracting, multiplying and dividing terms, including squares and cubes (8Ae.02)
Expand a bracket multiplied by a single term (including answers with squares and cubes), and factorise an expression by taking out the highest common factor (8Ae.03)

Algebra is arithmetic with letters. Every rule you use with numbers — brackets first, multiplication before addition, a × b = b × a — works in exactly the same way with letters. This page teaches you to read algebra correctly, simplify it, multiply out brackets (expand) and put brackets back in (factorise).

Key words

Word / phrase	Meaning / use
term	حد — a number, a letter, or numbers and letters multiplied together, e.g. 5, x, 3ab, 4y²
coefficient	معامل — the number in front of the letter; in 7x² the coefficient is 7
like terms	حدود متشابهة — terms with exactly the same letters and powers, e.g. 3x and −5x; 2a² and a²
expression	مقدار جبري — terms joined by + or −, with no equals sign, e.g. 3n + 5
formula	قانون / صيغة — a rule linking quantities, e.g. A = lw
equation	معادلة — two sides that are equal, with an unknown to find, e.g. 3n + 5 = 20
variable / unknown	متغير / مجهول — a letter that can change / a letter with a particular value to find
simplify	يبسّط — write in the shortest form by collecting like terms
expand	يفك الأقواس — multiply out the brackets
factorise	يحلل — write as a product by taking out a common factor
highest common factor (HCF)	العامل المشترك الأكبر — the biggest factor shared by all the terms

1. Expressions, formulae and equations

	Example	What the letter means	How to spot it
Expression	3n + 5	n can be any number (a variable). Different n gives a different value.	No equals sign
Formula	A = l × w	Each letter is a quantity (area, length, width). Change l or w and A changes.	Equals sign; letters on both sides stand for named quantities; one letter is the subject
Equation	3n + 5 = 20	n is an unknown with a particular value. Here n = 5 only.	Equals sign; you can solve it to find the letter

The same letter can play different roles. In the expression 2t + 1, t can be anything. In the formula s = d/t, t means 'time taken'. In the equation 2t + 1 = 9, t must be 4. Checkpoint may ask you to sort examples into expression / formula / equation, or to say what a letter represents.

2. Writing algebra the standard way

Instead of…	Write	Note
3 × a	3a	number first, no × sign
a × b	ab	letters usually in alphabetical order
1 × x	x	never write 1x
a × a	a²	a squared
a × a × a	a³	a cubed
y ÷ 4	y/4	division is written as a fraction
4 × (x + 2)	4(x + 2)	the number touches the bracket

Order of operations still applies. 3x² means 3 × x², so square x first, then multiply by 3. (3x)² means work out 3x first, then square. If x = −2: 3x² = 3 × (−2)² = 3 × 4 = 12, but (3x)² = (−6)² = 36. And x³ + 4 = (−2)³ + 4 = −8 + 4 = −4.

The laws of arithmetic work with letters: a + b = b + a and ab = ba (commutative); (a + b) + c = a + (b + c) and (ab)c = a(bc) (associative); a(b + c) = ab + ac (distributive). These laws are why we can collect like terms and expand brackets.

3. Collecting like terms

You can only add or subtract like terms: same letters, same powers. Each term owns the sign in front of it. Underline or colour each group of like terms.

Expression	Like terms grouped	Simplified
4a + 3b − a + 5b	(4a − a) + (3b + 5b)	3a + 8b
2x² + 5x − x² + 3x	(2x² − x²) + (5x + 3x)	x² + 8x
3ab + 2ba − ab	ab and ba are like terms	4ab
7y − 3 − 2y + 8	(7y − 2y) + (−3 + 8)	5y + 5
x² + x	not like terms	x² + x (cannot be simplified)

x² and x are not like terms (x² = x × x is a different thing from x), so x² + x stays as it is. Also x + x = 2x, but x × x = x².

4. Multiplying and dividing terms

Multiply the numbers, then multiply the letters. Same letters multiplied make a power: x × x = x², x × x² = x³. Divide by writing it as a fraction and cancelling numbers and letters.

Calculation	Numbers	Letters	Answer
3a × 4b	3 × 4 = 12	a × b = ab	12ab
2x × 5x	2 × 5 = 10	x × x = x²	10x²
3y × y²	3 × 1 = 3	y × y² = y³	3y³
4p × 2p × p	4 × 2 × 1 = 8	p × p × p = p³	8p³
12ab ÷ 3a	12 ÷ 3 = 4	ab ÷ a = b	4b
10x² ÷ 2x	10 ÷ 2 = 5	x² ÷ x = x	5x
18y³ ÷ 6y	18 ÷ 6 = 3	y³ ÷ y = y²	3y²

5. Expanding brackets

To expand a(b + c), multiply every term inside the bracket by the term outside: a(b + c) = ab + ac. An area diagram shows why.

A rectangle x by (x + 3) splits into two parts with areas x² and 3x, so x(x + 3) = x² + 3x.

Expand	Working	Answer
4(2x − 3)	4 × 2x = 8x; 4 × (−3) = −12	8x − 12
x(x + 7)	x × x = x²; x × 7 = 7x	x² + 7x
3a(2a − 5)	3a × 2a = 6a²; 3a × (−5) = −15a	6a² − 15a
y(y² + 4)	y × y² = y³; y × 4 = 4y	y³ + 4y
2p(p² − 3p)	2p × p² = 2p³; 2p × (−3p) = −6p²	2p³ − 6p²
−2(x − 4)	−2 × x = −2x; −2 × (−4) = +8	−2x + 8

Expand and simplify: expand each bracket, then collect like terms. 3(x + 2) + 2(x − 5) = 3x + 6 + 2x − 10 = 5x − 4. Watch the sign when a negative number is outside the bracket: 5(y + 1) − 2(y − 3) = 5y + 5 − 2y + 6 = 3y + 11.

فك الأقواس

اضرب الحد اللي برّه القوس في كل حد جوه القوس — كلهم، مش الأول بس. وخلي بالك من الإشارة: سالب × سالب = موجب، يعني −2(x − 4) = −2x + 8. وتذكّر: x × x = x² و x × x² = x³.

6. Factorising (taking out the HCF)

Factorising is the reverse of expanding: you put the brackets back in.

Expanding and factorising are inverse processes.

Find the HCF of the numbers (coefficients).
Find the HCF of the letters: a letter is in the HCF only if it is in every term; use its lowest power.
Write the HCF outside the bracket. Inside, write what each term must be multiplied by to give the original term (divide each term by the HCF).
Check by expanding. 'Factorise fully' means you must take out the highest common factor.

Expression	HCF	Factorised fully
6x + 9	3	3(2x + 3)
x² + 5x	x	x(x + 5)
8a² − 12a	4a	4a(2a − 3)
10y³ + 15y²	5y²	5y²(2y + 3)
12pq + 18p	6p	6p(2q + 3)
7m + 7	7	7(m + 1) — the 1 is needed!

8a² − 12a = 2a(4a − 6) is factorised, but not fully: 4a − 6 still has a common factor 2. The full answer is 4a(2a − 3).

Worked examples

Worked example 1 — Write E (expression), F (formula) or Q (equation) for each. (a) 5y − 2 (b) 5y − 2 = 18 (c) P = 4s, where P is the perimeter of a square of side s. [2]

(a) Expression — no equals sign; y can be any number.
(b) Equation — it can be solved: 5y = 20, so y = 4.
(c) Formula — it links two quantities (perimeter and side length); P changes when s changes.

Worked example 2 — Simplify (a) 7m + 3n − 2m − 5n (b) 4x² + 3x − x² − 7x [2]

(a) m terms: 7m − 2m = 5m. n terms: 3n − 5n = −2n. Answer 5m − 2n.
(b) x² terms: 4x² − x² = 3x². x terms: 3x − 7x = −4x. Answer 3x² − 4x.

Common mistake: combining x² and x terms (e.g. writing −x³ or −x² as the answer to (b)). They are not like terms.

Worked example 3 — Simplify (a) 5a × 3a (b) 2b × 3b² (c) 20c³ ÷ 4c [3]

(a) 5 × 3 = 15; a × a = a² → 15a².
(b) 2 × 3 = 6; b × b² = b × b × b = b³ → 6b³.
(c) 20 ÷ 4 = 5; c³ ÷ c = (c × c × c) ÷ c = c² → 5c².

Worked example 4 — x = −3. Work out the value of (a) 2x² (b) (2x)² (c) x³ + 10 [3]

(a) Indices before multiplication: x² = (−3)² = 9, then 2 × 9 = 18.
(b) Brackets first: 2x = −6, then (−6)² = 36.
(c) x³ = (−3) × (−3) × (−3) = −27, then −27 + 10 = −17.

Common mistake: in (a) working out (2 × −3)² = 36. Only x is squared in 2x².

Worked example 5 — Expand and simplify 3(2y + 5) − 2(y − 4). [2]

First bracket: 3 × 2y = 6y, 3 × 5 = 15 → 6y + 15.
Second bracket (the −2 goes with it): −2 × y = −2y, −2 × (−4) = +8 → −2y + 8.
Collect: 6y − 2y + 15 + 8 = 4y + 23.

Common mistake: writing −2(y − 4) = −2y − 8. A negative times a negative is positive.

Worked example 6 — (a) Expand a(a² + 3a). (b) Factorise fully 12x² − 18x. [3]

(a) a × a² = a³ and a × 3a = 3a² → a³ + 3a².
(b) HCF of 12 and 18 is 6. Both terms contain x; the lowest power is x. HCF = 6x.
12x² ÷ 6x = 2x and 18x ÷ 6x = 3 → 6x(2x − 3). Check: 6x × 2x − 6x × 3 = 12x² − 18x ✓

Worked example 7 — Kamal factorises 15n² + 10n and writes 5(3n² + 2n). Is his answer fully factorised? Explain. [1]

No. 3n² and 2n still have a common factor of n, so the HCF of the original terms is 5n, not 5.
Fully factorised: 5n(3n + 2).

Common mistakes

Adding unlike terms: 3x + 2y ≠ 5xy and x² + x ≠ x³.
Mixing up 2x (= x + x) and x² (= x × x).
Multiplying only the first term when expanding: 4(2x − 3) is 8x − 12, not 8x − 3.
Losing the sign: −2(y − 4) = −2y + 8.
Squaring the coefficient by mistake: 3x² with x = 4 is 3 × 16 = 48, not 12² = 144.
Not factorising fully: 8a² − 12a = 4a(2a − 3), not 2(4a² − 6a).
Forgetting the 1 when a whole term is the HCF: 7m + 7 = 7(m + 1), not 7m.

How Checkpoint asks this

'Simplify', 'Expand', 'Expand and simplify', 'Factorise fully' — each has a different meaning; do exactly what the command word says.
'Draw a line from each item to expression, formula or equation' or 'Tick the correct box'.
'Find the value of … when x = −3' — substitute with brackets around negative numbers, and follow the order of operations.
Area of a rectangle with sides like 2x and (x + 5): 'Write an expression for the area. Expand the brackets.'
'Ali says… Is he correct?' — check by expanding or substituting a number. Substituting a number (e.g. x = 2) into both forms is a quick way to test whether two expressions are equal.
Your final answer should be fully simplified; an unsimplified correct expression may lose the last mark.

Practice

Remember

Expression: no equals sign, letter = any number. Formula: links quantities. Equation: letter = unknown to find.
Only like terms (same letters, same powers) can be added or subtracted.
Multiply terms: numbers × numbers, letters × letters (x × x = x², x × x² = x³). Divide by cancelling.
Expand: multiply every term inside the bracket by the term outside — watch the signs.
Factorise fully: take out the HCF of the numbers AND the letters; check by expanding.

All pages in this course

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Stage 8 Maths: The Tests and How to Prepare

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Algebra

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Statistics and probability

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