Expanding Brackets and Factorising

Expanding means removing brackets by multiplying. Factorising is the reverse — putting expressions into brackets.

Expanding Single Brackets

Multiply each term inside the bracket by the term outside.

$$3(x + 4) = 3x + 12$$

$$-2(y - 5) = -2y + 10$$

Watch out: A negative outside changes signs inside! $$-(a + b) = -a - b$$

Expanding Double Brackets

There are two main methods to expand double brackets. You can use whichever you find clearer.

Method 1: FOIL

Use the acronym FOIL: First, Outer, Inner, Last.

$$(x + 3)(x + 5)$$

• First: $x \times x = x^2$
• Outer: $x \times 5 = 5x$
• Inner: $3 \times x = 3x$
• Last: $3 \times 5 = 15$

Combine: $x^2 + 5x + 3x + 15 = x^2 + 8x + 15$

Method 2: Distributing Terms (Splitting the First Bracket)

This method involves treating the second bracket as a single item. You split the first bracket into separate terms and multiply each one by the entire second bracket. This is very useful because it works for more complex problems (like trinomials) where FOIL does not.

Step 1: Break the first bracket $(x + 3)$ into $x$ and $+3$.

Step 2: Write each term multiplying the second bracket $(x + 5)$.

$$(x + 3)(x + 5) = x(x + 5) + 3(x + 5)$$

Step 3: Expand the single brackets.

$$= x^2 + 5x + 3x + 15$$

Step 4: Collect like terms.

$$= x^2 + 8x + 15$$

Visualizing the Split:

Factorising — Common Factor

Take out the highest common factor (HCF).

$$6x + 9 = 3(2x + 3)$$

$$x^2 + 5x = x(x + 5)$$

Factorising Quadratics (Higher)

For $x^2 + bx + c$, find two numbers that multiply to give $c$ and add to give $b$.

$$x^2 + 7x + 12 = (x + 3)(x + 4)$$

(since $3 \times 4 = 12$ and $3 + 4 = 7$)

Difference of Two Squares (Higher)

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

Example: $x^2 - 25 = (x + 5)(x - 5)$