Cambridge Syllabus Notes

Expressions & Symbolic Language

An expression is a collection of numbers, variables and operators that represents a value. Think of it as a recipe: the ingredients (numbers & variables) and the instructions (operators) tell you how to make a result.

Example 1: $3x + 5$ – 3 cups of variable x plus 5 whole units.
Example 2: $2a^2 - 4a + 6$ – a quadratic “dish” with a, a² and constants.

Exam Tip: Always look for common factors or patterns before simplifying. Quick checks can save time on the paper! 🚀

Simplifying Expressions

Combine like terms: $4x + 7x = 11x$.
Use distributive property: $3(x + 4) = 3x + 12$.
Reduce fractions: $\frac{6x}{3} = 2x$.

Formulae & Patterns

Formulae are like cheat‑codes that let you solve problems instantly. Memorise the most common ones:

Formula	Use
$a^2 - b^2 = (a+b)(a-b)$	Difference of squares
$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$	Sum of cubes
$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$	Difference of cubes

Exam Tip: Write the formula on a separate sheet and bring it into the exam. Quick reference saves time! 📄

Expansion (Distributive Property)

Expanding is like opening a gift box: you reveal all the hidden parts inside.

$$ (x + 3)(x - 2) = x(x-2) + 3(x-2) = x^2 - 2x + 3x - 6 = x^2 + x - 6 $$

First, multiply the first terms: $x \times x = x^2$.
Then the outer terms: $x \times (-2) = -2x$.
Next the inner terms: $3 \times x = 3x$.
Finally the last terms: $3 \times (-2) = -6$.

FOIL Method

FOIL stands for First, Outer, Inner, Last – a handy mnemonic for multiplying binomials.

Factorisation (Reversing Expansion)

Factorising is like putting the parts back together into a neat box.

$$ x^2 + x - 6 = (x + 3)(x - 2) $$

Look for two numbers that multiply to the constant term (-6) and add to the coefficient of the middle term (1). Those numbers are 3 and -2.
Rewrite the middle term using these numbers: $x^2 + 3x - 2x - 6$.
Group: $(x^2 + 3x) + (-2x - 6)$.
Factor each group: $x(x + 3) - 2(x + 3)$.
Factor out the common binomial: $(x + 3)(x - 2)$.

Exam Tip: When you see a quadratic, always try to factorise first. If it doesn't factorise nicely, check if it’s a perfect square or use the quadratic formula. 🧩

Perfect Square Trinomials

These look like $(a \pm b)^2$:

$a^2 + 2ab + b^2 = (a + b)^2$
$a^2 - 2ab + b^2 = (a - b)^2$

Example: $x^2 + 6x + 9 = (x + 3)^2$ because $2ab = 6x$ and $b^2 = 9$.

Quick Reference Cheat Sheet

Operation	Key Formula
Difference of Squares	$a^2 - b^2 = (a+b)(a-b)$
Sum of Cubes	$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
Difference of Cubes	$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$
Perfect Square	$a^2 \pm 2ab + b^2 = (a \pm b)^2$

Final Exam Reminder: Read the question carefully, write down what you know, and check your work by plugging the answer back into the original expression. Good luck! 🍀

Algebraic manipulation: expressions, formulae, expansion, factorisation