Edexcel International GCSE Mathematics A 4MA1: Use of symbols and algebraic manipulation – Revision Notes

For full explanations and worked examples, use the study guide for 2.1 and 2.2. These notes are for the final weeks.

They cover sections 2.1 Use of symbols and 2.2 Algebraic manipulation of Topic 2, Equations, formulae and identities, in the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), for the January and June series examined on it. Both tiers (Foundation papers 1F and 2F, Higher papers 1H and 2H) have content here; the Higher Tier statements are marked Higher tier only. A calculator may be used on every paper, but calculators with symbolic algebra are banned, so all manipulation is by hand.

Links: course hub · printable checklist · practice questions for 2.1 and 2.2 · earlier topic-level revision notes and practice set · free diagnostics

Key words (2.1A, 2.1B)

Word	Meaning	Example
Term	A number, letter or product of them	5x²y
Expression	Terms joined by + or −, no equals sign	3a − 2b + 7
Equation	True only for certain values of the letter	2x + 5 = 11
Formula	A rule linking variables	P = 2(l + w)
Variable	A letter that can take many values	r in A = πr²
Coefficient	The number multiplying a letter	−4 in −4x
Like terms	Same letters, same powers	3ab and −ab

Algebra obeys the rules of arithmetic: a + b = b + a, ab = ba, a(b + c) = ab + ac. Subtraction and division do not swap: a − b ≠ b − a in general.

Indices

Integer powers (both tiers, 2.1C, 2.1D)

Rule	Formula
Multiply	aᵐ × aⁿ = aᵐ⁺ⁿ
Divide	aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power of a power	(aᵐ)ⁿ = aᵐⁿ
Zero	a⁰ = 1
Negative	a⁻ⁿ = 1/aⁿ

The laws need the same base. With numbers in front, multiply the numbers and add the powers: 4x³ × 2x⁻¹ = 8x².

Fractional powers – Higher tier only (2.1A)

Form	Meaning	Example
a^(1/n)	nth root of a	27^(1/3) = 3
a^(m/n)	(nth root of a)ᵐ	27^(2/3) = 3² = 9
a^(−m/n)	1 ÷ a^(m/n)	4^(−1/2) = 1/2

Method: evaluating a^(−m/n)

1. Negative sign: take the reciprocal (flip a fraction).
2. Denominator n: take the nth root.
3. Numerator m: raise to the power m.

Substitution (2.2A)

Method

1. Write the expression with each letter replaced by its value in brackets.
2. Powers first, then × and ÷, then + and −.
3. Watch signs: (−3)² = 9, but −3² = −9.

Worked reminder: find (a − 2b)/c when a = 3.5, b = −1.25 and c = 4.

(3.5 - 2(-1.25)) / 4 = (3.5 + 2.5) / 4 = 6 / 4 = 1.5

Subtracting a negative adds. Work out the whole numerator before dividing, because the fraction line acts as a bracket.

Brackets and factors (2.2B to 2.2F)

Task	Reminder	Example
Collect like terms	Only add terms with identical letters and powers	4ab + 2a − ab = 3ab + 2a
Single bracket	Multiply every term inside	−2y(3y − 5) = −6y² + 10y
Common factor	Take out the HCF of numbers and letters	14x²y + 21xy = 7xy(2x + 3)
Two brackets	Each term × each term (4 products)	(x + 5)(x − 5) = x² − 25
x² + bx + c	Two numbers: product c, sum b	x² − x − 12 = (x − 4)(x + 3)

Sign patterns for x² + bx + c

c	b	Numbers
positive	positive	both positive
positive	negative	both negative
negative	either	one of each; the larger has the sign of b

Higher tier only (2.2A to 2.2E)

Three or more brackets

Method

1. Expand any two brackets and simplify.
2. Multiply the result by the next bracket (6 products for a quadratic × linear).
3. Collect like terms, highest power first.
4. Check with x = 1 or x = 2 in both forms.

Factorising ax² + bx + c

Method

1. Take out any common factor first.
2. Find two numbers with product ac and sum b.
3. Split bx using those numbers.
4. Factorise each pair; the brackets must match.
5. Expand to check.

Difference of two squares: p² − q² = (p − q)(p + q). It has no middle term.

Algebraic fractions

Task	Steps
Simplify	Factorise top and bottom fully, cancel whole brackets
Add or subtract	Common denominator, put each numerator in brackets, expand, collect
Multiply	Factorise everything, cancel, multiply tops and bottoms
Divide	Flip the second fraction, then multiply

Numeric denominators work the same way: use the lowest common multiple (for 4 and 6, use 12).

Completing the square

x² + bx + c = (x + b/2)² − (b/2)² + c

a(x² + …): take a out of the x² and x terms only, complete the square inside, then multiply the subtracted number by a.

Small reminder: x² + 6x − 2 = (x + 3)² − 9 − 2 = (x + 3)² − 11.

Proof

Object	Write as (n an integer)
Even number	2n
Odd number	2n + 1
Consecutive integers	n, n + 1
Consecutive even numbers	2n, 2n + 2
Multiple of k	k × (integer)

End with a sentence: “… = 4(n + 1), which is a multiple of 4 because n + 1 is an integer.”

Worked reminders (Higher tier only)

These are short, so you can redo them from memory before an exam.

Numeric denominators. Write (2x − 1)/5 + (x + 3)/2 as a single fraction.

LCM of 5 and 2 is 10
[2(2x - 1) + 5(x + 3)] / 10
= (4x - 2 + 5x + 15) / 10
= (9x + 13) / 10

Dividing. Simplify (x + 4)/(x² − 1) ÷ (2x + 8)/(x + 1).

flip and multiply:  (x + 4)/[(x - 1)(x + 1)] × (x + 1)/[2(x + 4)]
cancel (x + 4) and (x + 1):  1 / [2(x - 1)]

Choosing a method. Before factorising any quadratic, ask in this order: is there a common factor? Is it a difference of two squares? Is the x² coefficient 1? Each “yes” points to a quicker route than splitting the middle term.

Must-know distinctions

• Expand vs factorise. Expand removes brackets; factorise puts them in. They undo each other.
• x² + 25 vs x² − 25. Only the difference factorises: (x − 5)(x + 5). A sum of squares does not.
• (x + 5)² vs x² + 25. (x + 5)² = x² + 10x + 25.
• a⁻² vs −a². a⁻² = 1/a² is positive when a ≠ 0; −a² is never positive.
• Factor vs term. In (x + 2)(x − 1), (x + 2) is a factor and can cancel. In x + 2, the x is a term and cannot.
• Partly vs fully factorised. 4(3x² − 6x) is not fully factorised; 12x(x − 2) is.

Quick self-test

1. Simplify a⁻³ × a⁸.
2. Write down the value of 9⁰ and of 9⁻¹.
3. (Higher) Work out 16^(−3/4).
4. Find the value of 2x² − 5x when x = −3.
5. Factorise fully 12pq − 18q².
6. Expand and simplify (2y − 5)(y + 4).
7. Factorise x² + x − 42.
8. (Higher) Factorise 6x² + 7x − 3.
9. (Higher) Simplify (x² − 6x + 9)/(x² − 9).
10. (Higher) Write x² + 7x + 2 in the form (x + p)² + q.
11. (Higher) Expand and simplify (x − 2)³.
12. (Higher) Prove that (2n + 1)² − 1 is a multiple of 4 for every integer n.

Answers

1. a⁻³⁺⁸ = a⁵
2. 9⁰ = 1 and 9⁻¹ = 1/9
3. Reciprocal 1/16^(3/4); fourth root of 16 is 2; 2³ = 8; answer 1/8
4. 2(−3)² − 5(−3) = 18 + 15 = 33
5. 6q(2p − 3q)
6. 2y² + 8y − 5y − 20 = 2y² + 3y − 20
7. Product −42, sum +1: +7 and −6, so (x + 7)(x − 6)
8. ac = −18, sum 7: 9 and −2. 6x² + 9x − 2x − 3 = 3x(2x + 3) − (2x + 3) = (3x − 1)(2x + 3)
9. (x − 3)(x − 3) / [(x − 3)(x + 3)] = (x − 3)/(x + 3)
10. (x + 3.5)² − 12.25 + 2 = (x + 3.5)² − 10.25
11. (x − 2)² = x² − 4x + 4; × (x − 2) gives x³ − 6x² + 12x − 8
12. 4n² + 4n + 1 − 1 = 4n² + 4n = 4n(n + 1), which is 4 × an integer, so it is a multiple of 4.

Where marks are usually lost

• Writing x³ × x⁴ = x¹² (multiplying powers instead of adding them).
• Giving 5⁰ = 0 or 5⁻² = −25.
• Substituting a negative number without brackets, so −3² is worked out as −9 instead of 9.
• Stopping at a partial factorisation such as 3(4x²y − 6xy²) when the question says “factorise fully”.
• Dropping the middle term when squaring a bracket: (x − 6)² written as x² + 36 or x² − 36.
• Factorising ax² + bx + c with brackets that expand to the right x² and constant terms but the wrong middle term. Always expand to check.
• Cancelling terms instead of factors in algebraic fractions, for example crossing out x² in (x² + 3x)/(x² − 9).
• Forgetting to subtract every term of the second numerator: −(2x − 5) must become −2x + 5.
• Completing the square on 2x² − 8x + 1 as 2(x − 2)² − 4 + 1. The −4 must be multiplied by 2, giving 2(x − 2)² − 7.
• In a proof, checking a few numbers and calling it proved, or showing the algebra without the concluding sentence.

Official syllabus

Pearson Edexcel International GCSE in Mathematics (Specification A) (4MA1), Specification, Issue 2, November 2017, Pearson Education Limited (first assessment June 2018). Topic 2, Equations, formulae and identities: sections 2.1 Use of symbols and 2.2 Algebraic manipulation (Foundation and Higher Tier content).