Expanding And Factorising Brackets

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Notes

Expanding & Simplifying Single Brackets

To expand a bracket, multiply the term outside by **each term inside**.
Example: $3x(x + 2) = 3x \times x + 3x \times 2 = 3x^{2} + 6x$ .
Beware of minus signs: − × − = +, − × + = −.
When simplifying expressions with multiple brackets, expand each bracket first, then **collect like terms**.
Example: $2(x + 5) + 3x(x - 8) = 2x + 10 + 3x^{2} - 24x = 3x^{2} - 22x + 10$ .

Expanding Double Brackets

Multiply **every term** in the first bracket by **every term** in the second bracket (4 multiplications).
Use **FOIL** (First, Outer, Inner, Last) to remember the order.
A **grid** can help organise multiplication: write one bracket as row headings, the other as column headings, multiply cells, then sum.
Example: $(x + 1)(x + 3) = x^{2} + 3x + x + 3 = x^{2} + 4x + 3$ .
For squared brackets, rewrite as a product: $(x + 3)^{2} = (x + 3)(x + 3) = x^{2} + 6x + 9$ .
When expanding with multiple variables, combine only **like terms** (e.g., xy terms).

Expanding Triple Brackets

First expand and simplify **any two** brackets, then multiply the result by the third bracket.
Use a grid to multiply the resulting quadratic by the linear bracket.
Example: $(2x - 3)(x + 4)(3x - 1)$ → first expand $(2x - 3)(x + 4) = 2x^{2} + 5x - 12$ , then multiply by (3x − 1) to get $6x^{3} + 13x^{2} - 41x + 12$ .

Factorising Out Terms

Factorisation is the **reverse** of expanding brackets: write an expression as a product of factors.
Identify the **highest common factor (HCF)** of the coefficients and variables.
Write the HCF outside brackets and the remaining terms inside.
Example: $12x^{2} + 18x = 6x(2x + 3)$ .
Always factorise **fully** $(e.g., 2x(3x + 5)$ is not fully factorised; $6x(2x + 3) is)$ .
Check your answer by expanding the brackets.

Factorising by Grouping

Used when an expression has **four terms** and can be grouped into pairs with common factors.
Factorise each pair separately, then look for a **common bracket**.
Example: $xy + 3x + 5y + 15 = x(y + 3) + 5(y + 3) = (y + 3)(x + 5)$ .
The order of terms can be rearranged as long as grouping is possible.

Factorising Simple Quadratics (a = 1)

For $x^{2} + bx + c$ , find two numbers that **multiply to c** and **add to b**.
Write these numbers in brackets: $(x + p)(x + q)$ .
Example: $x^{2} - 2x - 8$ : numbers 2 and −4 → $(x + 2)(x - 4)$ .
Methods: inspection (quickest), splitting the middle term, or using a grid.

Factorising Harder Quadratics (a ≠ 1)

For $ax^{2} + bx + c$ , find two numbers that **multiply to ac** and **add to b**.
Use these numbers to **split the middle term**, then factorise by grouping.
Example: $4x^{2} - 25x - 21$ : $ac = -84, b = -25$ → numbers −28 and 3 → $4x^{2} - 28x + 3x - 21 = 4x(x - 7) + 3(x - 7) = (x - 7)(4x + 3)$ .
A grid can also be used with the split terms.

Difference of Two Squares

The **difference of two squares** is $a^{2} - b^{2} = (a + b)(a - b)$ .
Both terms must be perfect squares and subtracted.
Example: $9x^{2} - 16 = (3x)^{2} - 4^{2} = (3x + 4)(3x - 4)$ .
Can be applied to powers: r⁸ − t⁶ = (r⁴)² $- (t^{3})^{2} =$ (r⁴ + t³)(r⁴ $- t^{3})$ .
Sometimes a common factor must be taken out first: $2y^{2} - 50 = 2(y^{2} - 25) = 2(y + 5)(y - 5)$ .

Deciding the Factorisation Method

For **two terms**: check for common factor or difference of two squares.
For **three terms** (quadratic): if $a = 1$ , use simple factorisation; if $a \ne 1$ , check for a common factor first, then use grouping or grid.
If the quadratic has a common factor, factor it out first: $3x^{2} + 15x + 18 = 3(x^{2} + 5x + 6) = 3(x + 2)(x + 3)$ .
Check if $b^{2} - 4ac$ is a perfect square to see if the quadratic factorises.
Always factorise **fully** and check by expanding.

Expanding Double Brackets Using a Grid

Factorising by Grouping Example

Difference of Two Squares Visual

Expanding Triple Brackets Steps

Practice questions

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1.Expand 7(x – 8).
Easy
- A7x – 56
- B7x – 8
- C $7x + 56$
- Dx – 56
2.Factorise $5p + pt$ .
Easy
- A $p(5 + t)$
- B $5(p + t)$
- Cp(5t)
- D $5p(1 + t)$
3.Factorise $12x + 15$ .
Easy
- A $3(4x + 5)$
- B $12(x + 15)$
- C $3(4x + 15)$
- D $12x(1 + 15x)$
4.Factorise 5y – 6py.
Easy
- Ay(5 – 6p)
- B5y(1 – 6p)
- Cy(5 – 6)
- D5(1 – 6p)
5.Factorise $2x^{2}$ – x.
Easy
- Ax(2x – 1)
- B2x(x – 1)
- Cx(2x – x)
- D $2x^{2}(1$ – 1/2x)
6.Factorise completely $21a^{2} + 28ab$ .
Medium
- A $7a(3a + 4b)$
- B $7(3a^{2} + 4ab)$
- C $21a(a + 4b)$
- D $7a(3a + 4)$
7.Expand and simplify (m – $3)(m + 2)$ .
Medium
- A $m^{2}$ – m – 6
- B $m^{2}$ – 5m – 6
- C $m^{2}$ – $m + 6$
- D $m^{2} + m$ – 6
8.Expand and simplify $(x + 3)(x + 5)$ .
Medium
- A $x^{2} + 8x + 15$
- B $x^{2} + 8x + 8$
- C $x^{2} + 15x + 8$
- D $x^{2} + 2x + 15$

Summary

## Expanding & Factorising Brackets ### Expanding Single Brackets Multiply the term outside by each term inside. - **Example:** `3x(x + 2) = 3x² + 6x` - Beware of minus signs: `− × − = +`, `− × + = −`. - For multiple brackets, expand each then collect like terms. ### Expanding Double Brackets Every term in first bracket multiplies every term in second (FOIL: First, Outer, Inner, Last). - **Example:** `(x + 1)(x + 3) = x² + 4x + 3` - Use a grid for visual aid. - For squared brackets: `(x + 3)² = (x + 3)(x + 3) = x² + 6x + 9` (not `x² + 9`). ### Expanding Triple Brackets Expand two brackets first, then multiply by the third. - **Example:** `(2x - 3)(x + 4)(3x - 1)` → expand first two: `2x² + 5x - 12`, then multiply by `(3x - 1)` to get `6x³ + 13x² - 41x + 12`. ### Factorising Out Terms Reverse of expanding: find HCF of terms and place outside brackets. - **Example:** `12x² + 18x = 6x(2x + 3)` - Factorise **fully**: `x(6x + 10)` is not fully factorised; `2x(3x + 5)` is. ### Factorising by Grouping Group terms with common factors, factor each group, then factor out common bracket. - **Example:** `xy + 3x + 5y + 15 = x(y + 3) + 5(y + 3) = (y + 3)(x + 5)` ### Factorising Simple Quadratics (`x² + bx + c`) Find two numbers that multiply to `c` and add to `b`. - **Example:** `x² - 2x - 8 = (x + 2)(x - 4)` because `2 × (-4) = -8` and `2 + (-4) = -2`. ### Factorising Harder Quadratics (`ax² + bx + c`, a ≠ 1) Find two numbers that multiply to `ac` and add to `b`. Split middle term, then factor by grouping or use a grid. - **Example:** `4x² - 25x - 21`: numbers `-28` and `+3` (since `4×(-21) = -84`, sum `-25`). Then `4x² - 28x + 3x - 21 = 4x(x - 7) + 3(x - 7) = (x - 7)(4x + 3)`. ### Difference of Two Squares `a² - b² = (a + b)(a - b)` - **Example:** `x² - 9 = (x + 3)(x - 3)` - Can involve coefficients: `4m² - 25n² = (2m + 5n)(2m - 5n)` - Factor out common factor first: `2x² - 18 = 2(x² - 9) = 2(x + 3)(x - 3)` ### Deciding the Factorisation Method - **2 terms:** Check for common factor or difference of two squares. - **3 terms with `x²`:** Simple quadratics (find two numbers). - **3 terms with `ax²`:** Check for common factor first; otherwise harder quadratics. - **Cubic:** Factor out `x` first, then factorise quadratic. **Always check by expanding your answer!**

Key terms

Expand: To multiply out brackets, removing them by multiplying each term inside by the term outside.
Factorise: To write an expression as a product of its factors (the reverse of expanding).
Highest Common Factor (HCF): The largest factor that divides all terms in an expression.
FOIL: A method for expanding double brackets: First, Outer, Inner, Last.
Quadratic: An expression of the form ax² + bx + c, where a ≠ 0.
Difference of Two Squares: A rule: a² - b² = (a + b)(a - b). Used to factorise expressions of the form (something)² - (something else)².
Factorising by Grouping: A method where terms are grouped, each group factorised, then a common bracket factorised out.

Slides

Expanding Single Brackets

To expand a bracket, multiply the term outside by each term inside.
For example, $3x(x + 2) = 3x \times x + 3x \times 2 = 3x^{2} + 6x$ .
Be careful with minus signs: $- \times - = + and - \times + = -$ .

Expand 4x(2x − 3).

$8x^{2} - 12x$
$8x^{2} + 12x$
$8x^{2} - 3$
$4x^{2} - 12x$

Expanding and Simplifying Multiple Brackets

When adding or subtracting brackets, expand each bracket separately first.
Then collect like terms (terms with the same variable and power).
Example: $2(x + 5) + 3x(x - 8) = 2x + 10 + 3x^{2} - 24x = 3x^{2} - 22x + 10$ .

Expand and simplify: $5(p + 3) - 2(1 - 2p) = ____ p + ____$

Expanding Double Brackets

Every term in the first bracket must multiply every term in the second bracket.
Use FOIL: First, Outer, Inner, Last.
Example: $(x + 1)(x + 3) = x^{2} + 3x + x + 3 = x^{2} + 4x + 3$ .

Expand $(x + 7)(x - 3)$ .

$x^{2} + 4x - 21$
$x^{2} - 4x - 21$
$x^{2} + 10x - 21$
$x^{2} + 4x + 21$

Expanding Squared Brackets

A bracket squared means multiplying the bracket by itself.
Write $(x + 3)^{2}$ as $(x + 3)(x + 3)$ and expand using FOIL.
Result: $x^{2} + 6x + 9$ . Do not write $x^{2} + 9$ .

Expand $(2x + 3)^{2} = ____ x^{2} + ____ x + ____$

Expanding Triple Brackets

First expand and simplify any two brackets, then multiply the result by the third bracket.
Use a grid to help keep track of terms when multiplying a quadratic by a linear bracket.
Example: $(x + 2)(x^{2} + 3x + 1) = x^{3} + 5x^{2} + 7x + 2$ .

Expand $(x - 2)(x + 5)(2x - 1)$ .

$2x^{3} + 5x^{2} - 23x + 10$
$2x^{3} + 5x^{2} - 23x - 10$
$2x^{3} + 5x^{2} + 23x + 10$
$2x^{3} - 5x^{2} - 23x + 10$

Factorising Out Terms

Factorisation is the reverse of expanding brackets.
Find the highest common factor (HCF) of the terms and place it outside brackets.
Example: $12x^{2} + 18x = 6x(2x + 3)$ . Check by expanding.

Factorise completely: $4 - 8x = ____ (____ - ____)$

Factorising by Grouping

Used when there are four terms and a common bracket can be formed.
Group terms in pairs, factorise each pair, then factor out the common bracket.
Example: $xy + 3x + 5y + 15 = x(y+3) + 5(y+3) = (y+3)(x+5)$ .

Factorise $ab + 3b + 2a + 6$ .

(a+3)(b+2)
(a+2)(b+3)
(a+3)(b+3)
(a+2)(b+2)

Factorising Simple Quadratics (a=1)

For $x^{2} + bx + c$ , find two numbers that multiply to c and add to b.
Write as $(x + p)(x + q)$ where p and q are those numbers.
Example: $x^{2} - 2x - 8 = (x + 2)(x - 4)$ because $2\times (-4)=-8$ and $2+(-4)=-2$ .

Factorise $x^{2} - 4x - 21 = (x + ____)(x - ____)$

Factorising Harder Quadratics (a≠1)

For $ax^{2} + bx + c$ , find two numbers that multiply to ac and add to b.
Split the middle term using those numbers, then factorise by grouping.
Example: $4x^{2} - 25x - 21: ac=-84$ , numbers −28 and 3. Rewrite as $4x^{2}-28x+3x-21$ , then factorise to (x−7)(4x+3).

Factorise $6x^{2} - 7x - 3$ .

(3x+1)(2x−3)
(3x−1)(2x+3)
(2x+1)(3x−3)
(2x−1)(3x+3)

Difference of Two Squares

$a^{2} - b^{2} = (a + b)(a - b)$ .
Both terms must be perfect squares separated by a minus sign.
Example: $9x^{2} - 16 = (3x)^{2} - 4^{2} = (3x+4)(3x-4)$ .

Factorise $4r^{2} -$ t⁴ = (____ + ____)(____ − ____)

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