Algebraic manipulation: expressions, formulae, expansion, factorisation

Cambridge IGCSE Mathematics 0580 – Algebraic Manipulation (Core + Extended)

1. How this section fits the syllabus

• Assessment Objectives
  • AO1 – Recall and apply definitions, facts and procedures.
  • AO2 – Solve problems by manipulating algebraic expressions, equations and formulae.
• This note covers every Core requirement for Papers 1, 2 and 3 and all Extended material needed for Paper 4 (differentiation) and the optional extended sections of Papers 1‑3.
• Key exam‑tips are highlighted in green.

2. Number foundations that support algebraic work (syllabus C1‑C1.18 / E1‑E1.18)

• Natural numbers, integers, prime numbers, squares, cubes, rational & irrational numbers.
• Standard form, estimation, bounds and rounding.
• Indices (including fractional and negative exponents) – see Section 4.
• Percentages, ratios, rates, direct & inverse proportion.
• Recurring decimals (Extended) and simple surds.

3. Algebraic expressions – the building blocks

• Term: part of an expression separated by + or – signs.
• Coefficient: numerical factor of a variable (e.g. 5 in 5x).
• Degree: highest power of the variable(s) in a term or in the whole expression.
• Standard form: arrange terms in descending order of degree.

Example: 4x² – 7xy + 3y² has three terms, coefficients 4, –7, 3, and overall degree 2.

4. Indices (powers) – rules you must know (C1.6‑C1.9 / E1.6‑E1.9)

Rule	Mathematical form	Example
Product rule	aⁿ·aᵐ = aⁿ⁺ᵐ	2³·2⁴ = 2⁷
Quotient rule	aⁿ / aᵐ = aⁿ⁻ᵐ	5⁶ / 5² = 5⁴
Power of a power	(aⁿ)ᵐ = aⁿᵐ	(3²)³ = 3⁶
Power of a product	(ab)ⁿ = aⁿbⁿ	(2x)³ = 8x³
Zero exponent	a⁰ = 1 (a≠0)	7⁰ = 1
Negative exponent	a⁻ⁿ = 1 / aⁿ	4⁻² = 1/16
Fractional exponent	a¹⁄ⁿ = ⁿ√a	27¹⁄³ = 3

Exam tip: Index rules are **not** supplied in the paper – you must memorise them.

5. Algebraic fractions (C2.4‑C2.6 / E2.4‑E2.6)

5.1 Simplifying

• Factor numerator and denominator, then cancel common factors.
• Write the final answer in lowest terms.

Example:
\[ \frac{6x^{2}+9x}{3x}= \frac{3x(2x+3)}{3x}=2x+3 \]

5.2 Adding / Subtracting

• Find the least common denominator (LCD).
• Rewrite each fraction with the LCD, then combine numerators.

Example:
\[ \frac{2}{x}+\frac{3}{x+1}= \frac{2(x+1)+3x}{x(x+1)}=\frac{5x+2}{x(x+1)} \]

5.3 Multiplying / Dividing

• Multiply straight across; for division, multiply by the reciprocal.

Example (division):
\[ \frac{5x}{2}\div\frac{x}{4}= \frac{5x}{2}\times\frac{4}{x}=10 \]

5.4 Inverting a formula

When a formula is given as a fraction, isolate the required variable by cross‑multiplication.

Example – make r the subject of \(\displaystyle A=\frac{\pi r^{2}}{h}\):
\[ A=\frac{\pi r^{2}}{h}\Longrightarrow Ah=\pi r^{2}\Longrightarrow r=\sqrt{\frac{Ah}{\pi}} \]

6. Manipulating formulae – making a variable the subject (C2.7 / E2.7)

1. Identify the target variable.
2. Use inverse operations to move all other terms to the opposite side – remember to change the sign when moving across the equals sign.
3. Remove brackets by expanding or using the distributive law in reverse.
4. If the variable appears in more than one term, factor it out.
5. Check by substituting a convenient number.

Example – isolate t in \(\displaystyle s=ut+\tfrac12 at^{2}\) (quadratic in t):
\[ \frac12 at^{2}+ut-s=0\Longrightarrow at^{2}+2ut-2s=0 \] Use the quadratic formula (provided in the exam):
\[ t=\frac{-2u\pm\sqrt{(2u)^{2}+8as}}{2a} =\frac{-u\pm\sqrt{u^{2}+2as}}{a} \]

Exam tip: The quadratic formula is supplied; you must know how to apply it correctly.

7. Expanding and factorising (C2.8‑C2.10 / E2.8‑E2.10)

7.1 Standard algebraic identities (must be memorised)

Identity	Expanded form
\((a+b)^{2}\)	\(a^{2}+2ab+b^{2}\)
\((a-b)^{2}\)	\(a^{2}-2ab+b^{2}\)
\((a+b)(a-b)\)	\(a^{2}-b^{2}\)
\((ax+by)(cx+dy)\)	\(acx^{2}+(ad+bc)xy+bdy^{2}\)

7.2 Factorising techniques

• Common factor (GCF) – factor out the greatest numerical and/or algebraic factor first.
• Difference of squares – \(a^{2}-b^{2}=(a+b)(a-b)\).
• Perfect square trinomials – \(a^{2}\pm2ab+b^{2}=(a\pm b)^{2}\).
• Quadratic factorisation (a = 1) – find two numbers that multiply to c and add to b.
• Quadratic factorisation (a ≠ 1) – use the “ac‑method” or factor by grouping.
• Completing the square – rewrite \(ax^{2}+bx+c\) as \(a(x-h)^{2}+k\).
• Higher‑degree factorisation – look for patterns (cubic sum/difference, grouping, substitution).

Example – factorise \(2x^{2}+7x+3\) (ac‑method):
\[ 2\cdot3=6;\; \text{numbers }1\text{ and }6\text{ give }1\cdot6=6,\;1+6=7. \] \[ 2x^{2}+x+6x+3=x(2x+1)+3(2x+1)=(2x+1)(x+3) \]

7.3 Completing the square (required for quadratic formula derivation and solving)

1. If \(aeq1\), divide the whole equation by \(a\).
2. Rewrite as \(x^{2}+\frac{b}{a}x = -\frac{c}{a}\).
3. Add \(\bigl(\frac{b}{2a}\bigr)^{2}\) to both sides.
4. Factor the left‑hand side as a perfect square and solve for \(x\).

Example: \(x^{2}+6x+5=0\) → \((x+3)^{2}=4\) → \(x=-3\pm2\) → \(x=-1\) or \(-5\).

8. Solving equations (C2.1‑C2.3 / E2.1‑E2.3)

8.1 Linear equations

• One‑variable: \(ax+b=0\) → \(x=-\dfrac{b}{a}\).
• Two‑variable simultaneous (non‑fractional): substitution or elimination.
• Fractional linear equations – first clear denominators.

Example (fractional): \(\displaystyle \frac{2}{x-1}=3\) → \(2=3(x-1)\) → \(x=\frac{5}{3}\).

8.2 Quadratic equations

Three exam‑relevant methods:

1. Factorisation (when possible).
2. Completing the square.
3. Quadratic formula (provided in the exam): \[ x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \]

8.3 Rational (fractional) equations

• Identify the LCD and multiply through to remove denominators.
• Solve the resulting polynomial.
• Check all solutions in the original equation – discard any that make a denominator zero.

8.4 Simultaneous linear equations (two variables)

Standard form:

\[ \begin{cases} a_{1}x+b_{1}y=c_{1}\\[2pt] a_{2}x+b_{2}y=c_{2} \end{cases} \]

• Elimination – add/subtract multiples to cancel one variable.
• Substitution – solve one equation for a variable, substitute into the other.
• Always check the solution in both original equations.

9. Inequalities (C2.5‑C2.6 / E2.5‑E2.6)

9.1 Linear inequalities (one variable)

• Treat like equations; reverse the inequality sign when multiplying/dividing by a negative number.
• Express the solution in interval notation or as a range.

Example: \(-3x+4>1\) → \(-3x>-3\) → \(x<1\).

9.2 Quadratic inequalities

1. Factor or use the quadratic formula to find the roots.
2. Mark the roots on a number line; test a value in each interval.
3. Include or exclude the roots depending on “≤” or “≥”.

Example: \(x^{2}-5x+6\le0\) → \((x-2)(x-3)\le0\) → solution \(2\le x\le3\).

9.3 Reciprocal and absolute‑value inequalities

• For \(\displaystyle \frac{1}{x}>k\) consider the sign of the denominator (split into two cases).
• For \(|ax+b|

10. Sequences & proportion (C3.1‑C3.4 / E3.1‑E3.4)

10.1 Arithmetic sequences

• General term: \(a_{n}=a_{1}+(n-1)d\).
• Sum of first \(n\) terms: \(S_{n}= \dfrac{n}{2}\bigl(2a_{1}+(n-1)d\bigr)\).

10.2 Geometric sequences

• General term: \(a_{n}=a_{1}r^{\,n-1}\).
• Finite sum: \(S_{n}=a_{1}\dfrac{1-r^{n}}{1-r}\) ( \(req1\) ).
• Infinite sum (|r| < 1): \(S_{\infty}= \dfrac{a_{1}}{1-r}\).

10.3 Direct & inverse proportion

• Direct: \(y=kx\) → \(k=\dfrac{y}{x}\).
• Inverse: \(y=\dfrac{k}{x}\) → \(k=xy\).

10.4 Cubic sequences (Extended)

When the second differences are not constant but the third differences are, the sequence is cubic. Use the method of finite differences to find the formula.

11. Graphs of functions (C4.1‑C4.4 / E4.1‑E4.4)

Function type	Standard form	Key features
Linear	y=mx+c	Slope m, y‑intercept c; straight line.
Quadratic	y=ax²+bx+c	Parabola; axis of symmetry \(x=-\dfrac{b}{2a}\); vertex \((h,k)\) where \(h=-\dfrac{b}{2a}\).
Reciprocal	y=\dfrac{k}{x}\)	Two hyperbolic branches; asymptotes x=0 and y=0.
Exponential	y=ab^{x}	Growth if b>1, decay if 0

• Solid line = part of the graph that satisfies the inequality (≤ or ≥).
  Broken line = part that does **not** satisfy the inequality (< or >).
• Shading indicates the solution region for two‑dimensional inequalities.

12. Coordinate geometry basics (C5.1‑C5.5 / E5.1‑E5.5)

• Cartesian coordinates – plotting points \((x,y)\).
• Gradient (slope): \(m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\).
• Equation of a straight line: \(y=mx+c\) or \(ax+by=c\).
• Parallel lines: equal gradients.
  Perpendicular lines: product of gradients = –1.
• Distance formula: \(d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\).
• Mid‑point formula: \(\displaystyle M\Bigl(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\Bigr)\).
• Extended: equation of a perpendicular bisector (useful for circle geometry).

13. Extended algebra topics (E2.11‑E2.13)

• Functions: domain, range, composite \((f\circ g)(x)\), and inverse \(f^{-1}(x)\).
• Differentiation (introductory) – for simple powers: \[ \frac{d}{dx}\bigl(x^{n}\bigr)=nx^{\,n-1} \] Used for finding gradients of curves and solving optimisation problems.
• Application of differentiation to quadratic and cubic functions (e.g., find the maximum/minimum of \(y=ax^{2}+bx+c\)).

14. General exam advice (green highlights)

• All index rules, algebraic identities and the quadratic formula must be memorised – they are **not** supplied.
• Check every answer in the original equation, especially after clearing denominators.
• When a formula is supplied, you only need to rearrange it; you are not expected to derive it.
• Use a calculator only where permitted (usually for numerical evaluation, not for algebraic manipulation).
• Write your working clearly and label each step – marks are awarded for method as well as final answer.