Expanding and Factorising Expressions Revision Notes | GCSE AQA Higher Maths

1. Why Expanding and Factorising Matter

Algebra looks like a foreign language until you crack the code. Expanding and factorising are the two fundamental skills that unlock every algebraic problem you'll face at GCSE.

Think of expanding as 'unpacking' brackets - you multiply everything out to see all the terms clearly. Factorising does the opposite - it 'packs' expressions back into brackets, which is essential for solving quadratic equations.

These skills appear in every GCSE paper. You'll use them to solve equations, simplify expressions, and tackle word problems. Master these techniques now, and algebra becomes much less intimidating.

2. The Essential Toolkit

The FOIL method gives you a foolproof system. Take $(x + 3)(x + 5)$ - multiply First terms ($x \times x = x^2$), Outer terms ($x \times 5 = 5x$), Inner terms ($3 \times x = 3x$), Last terms ($3 \times 5 = 15$).

The difference of two squares is your secret weapon. Expressions like $x^2 - 9$ or $4y^2 - 25$ factorise instantly using this pattern. Spot the two perfect squares, find their square roots, and write as $(\text{sum})(\text{difference})$.

3. Expanding Step by Step

Let's work through expanding systematically. The key is being methodical - rushing leads to sign errors and missed terms.

Worked example: Expanding Two Binomials

Expand $(2x + 3)(x - 4)$

1. $2x \times x = 2x^2$ — First term from each bracket
2. $2x \times (-4) = -8x$ — First term of first bracket × second term of second bracket
3. $3 \times x = 3x$ — Second term of first bracket × first term of second bracket
4. $3 \times (-4) = -12$ — Second term from each bracket
5. $2x^2 + (-8x) + 3x + (-12)$ — Combine all four products
6. $2x^2 - 5x - 12$ — Combine $-8x + 3x = -5x$

Answer: $2x^2 - 5x - 12$

Notice how we kept track of signs carefully. The minus sign in $(x - 4)$ affects both multiplications involving the $-4$. This is where most students make errors.

Worked example: Expanding Three Brackets

Expand $(x + 1)(x + 2)(x + 3)$

1. $(x + 1)(x + 2) = x^2 + 3x + 2$ — Use FOIL: $x^2 + 2x + x + 2 = x^2 + 3x + 2$
2. $(x^2 + 3x + 2)(x + 3)$ — Now we have a trinomial times a binomial
3. $x^2(x + 3) + 3x(x + 3) + 2(x + 3)$ — Each term in the first bracket multiplies the second bracket
4. $x^3 + 3x^2 + 3x^2 + 9x + 2x + 6$ — Multiply out each bracket separately
5. $x^3 + 6x^2 + 11x + 6$ — Combine $3x^2 + 3x^2 = 6x^2$ and $9x + 2x = 11x$

Answer: $x^3 + 6x^2 + 11x + 6$

With three brackets, expand two first, then multiply by the third. Don't try to do everything at once - that's a recipe for mistakes.
Tip: Use a coordinate grid to visually represent the expansion.

4. Factorising Techniques

Factorising reverses the expanding process. You're looking for brackets that multiply to give the original expression. This skill is crucial for solving quadratic equations.

Worked example: Factorising a Quadratic

Factorise $x^2 + 7x + 12$

1. $x^2 + 7x + 12$ has form $x^2 + bx + c$ — We need two numbers that multiply to 12 and add to 7
2. $1 \times 12$, $2 \times 6$, $3 \times 4$ — All possible ways to multiply to get 12
3. $1 + 12 = 13$, $2 + 6 = 8$, $3 + 4 = 7$ ✓ — Only 3 and 4 add to give the middle coefficient
4. $(x + 3)(x + 4)$ — Put the two numbers as the second terms in each bracket
5. $(x + 3)(x + 4) = x^2 + 7x + 12$ ✓ — Always verify your factorisation is correct

Answer: $(x + 3)(x + 4)$

The key insight is finding two numbers that multiply to the constant term and add to the coefficient of $x$. This systematic approach works for any quadratic of the form $x^2 + bx + c$.

Worked example: Difference of Two Squares

Factorise $9x^2 - 16$

1. $9x^2 - 16 = (3x)^2 - 4^2$ — Both terms are perfect squares being subtracted
2. $\sqrt{9x^2} = 3x$ and $\sqrt{16} = 4$ — Find what was squared to get each term
3. $a^2 - b^2 = (a + b)(a - b)$ — Use the difference of squares pattern
4. $(3x + 4)(3x - 4)$ — Replace $a$ with $3x$ and $b$ with $4$
5. $(3x + 4)(3x - 4) = 9x^2 - 16$ ✓ — Check: $9x^2 - 12x + 12x - 16 = 9x^2 - 16$

Answer: $(3x + 4)(3x - 4)$

The difference of two squares pattern is incredibly useful. Look for expressions where two perfect squares are subtracted - they factorise instantly using this method.

5. Exam Success and Common Traps

Expanding and factorising questions appear on every GCSE paper, often worth 3-4 marks each. The good news is they're very predictable - master the patterns and you'll score consistently.

Examiners love to test your attention to detail. Sign errors are the biggest trap - students often lose marks by mishandling negative signs when expanding or factorising.

Another common mistake is rushing the factorising process. Students guess factor pairs instead of systematically checking which ones work. Take time to verify your answer by expanding back.

The graph shows why factorising matters. The quadratic $y = x^2 + 5x + 6$ crosses the x-axis where $y = 0$, which means $x^2 + 5x + 6 = 0$. Factorising gives $(x + 2)(x + 3) = 0$, so $x = -2$ or $x = -3$ - exactly where the curve crosses the axis.