Expanding And Factorising Brackets

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Notes

Expanding Single Brackets

**Expanding** means multiplying the term outside the bracket by each term inside.
For example, **3x(x + 2)** $= 3x \times x + 3x \times 2 =$ **3x² + 6x**.
Watch out for **negative signs**: − × − = +, − × + = −.
Use brackets around negative terms to avoid mistakes.

Simplifying After Expanding Single Brackets

First expand each bracket separately, then **collect like terms**.
Example: $4(x + 7) + 5x(3 - x) = 4x + 28 + 15x - 5x^{2} =$ **19x + 28 − 5x²**.
Be careful with subtraction: $3x(x + 2) - 7(x - 6) = 3x^{2} + 6x - 7x + 42 =$ **3x² − x + 42**.

Expanding Double Brackets

Multiply **every term** in the first bracket by **every term** in the second.
Use **FOIL**: First, Outer, Inner, Last.
Example: $(x + 1)(x + 3) =$ x·x + x·3 + 1·x + 1·3 = **x² + 4x + 3**.
A **grid** can help organise the multiplication visually.

Expanding a Bracket Squared

Write $(x + 3)^{2}$ as **(x + 3)(x + 3)** and expand using FOIL.
Example: $(x + 3)^{2} = x^{2} + 3x + 3x + 9 =$ **x² + 6x + 9**.
Common mistake: $(x + y)^{2}$ is **not** $x^{2} + y^{2}$ .

Factorising Out Terms

**Factorising** is the reverse of expanding: write as a product of factors.
Find the **highest common factor (HCF)** of the number parts and algebra parts.
Place the HCF outside brackets and the remaining terms inside.
Example: $12x^{2} + 18x =$ **6x(2x + 3)**.

Factorising with Multiple Variables

Find the HCF of numbers and each variable separately.
Example: $2a^{3}b^{2} - 4a^{2}b^{3}$ : number $HCF = 2, a HCF = a^{2}, b HCF = b^{2}$ → overall $HCF =$ **2a²b²**.
Result: $2a^{2}b^{2}(a - 2b)$ .
Always check by expanding: factorise **fully** $(e.g., 2x(3x + 5)$ not $x(6x + 10))$ .

Exam Tips

Check factorisation by **expanding the brackets**.
**Factorise fully** means take out the greatest common factor.
Practice with past paper questions: easy, medium, hard.

Expanding Single Bracket: 3x(x + 2)

Grid Method for Double Brackets: (x + 1)(x + 3)

Factorising: 12x² + 18x = 6x(2x + 3)

FOIL Method for (2x − 3)(x + 4)

Practice questions

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1.Expand $5(x - 3)$ .
Easy
- A $5x - 15$
- B $5x - 3$
- C $5x + 15$
- D $5x + 3$
2.Factorise $3x + 12$ .
Easy
- A $3(x + 4)$
- B $3(x + 12)$
- C $x(3 + 12)$
- D $3x(1 + 4)$
3.Factorise $5y - 6py$ .
Easy
- A $y(5 - 6p)$
- B $y(5 - 6)$
- C $5y(1 - 6p)$
- D $p(5y - 6)$
4.Factorise $12x + 15$ .
Easy
- A $3(4x + 5)$
- B $3(4x + 15)$
- C $12(x + 15)$
- D $5(12x + 3)$
5.Factorise $5p + pt$ .
Easy
- A $p(5 + t)$
- B $5(p + t)$
- C $p(5 + pt)$
- D $t(5p + 1)$
6.Multiply out $9(3 - x)$ .
Easy
- A $27 - 9x$
- B $27 + 9x$
- C $9x - 27$
- D $27 - x$
7.Factorise completely $18x - 24$ .
Medium
- A $6(3x - 4)$
- B $3(6x - 8)$
- C $2(9x - 12)$
- D $18(x - 6)$
8.Expand $x(2y + x)$ .
Medium
- A $2xy + x^{2}$
- B $2xy + x$
- C $x^{2} + 2y$
- D $2x^{2}y$

Summary

## Expanding & Factorising Brackets ### Expanding Single Brackets - Multiply the term outside by **each term** inside. - Watch signs: $- \times - = +$, $- \times + = -$. - Example: $4x(2x-3) = 8x^2 - 12x$. ### Expanding & Simplifying - Expand each bracket separately, then **collect like terms**. - Example: $2(x+5) + 3x(x-8) = 2x+10+3x^2-24x = 3x^2 -22x +10$. ### Expanding Double Brackets - Use **FOIL** (First, Outer, Inner, Last) or a grid. - $(x+1)(x+3) = x^2 + 4x + 3$. - For squares: $(x+3)^2 = (x+3)(x+3) = x^2+6x+9$. - **Never** write $(x+y)^2 = x^2+y^2$. ### Factorising - Reverse of expanding: put common factor outside brackets. - Find **HCF** of numbers and variables. - Example: $12x^2+18x = 6x(2x+3)$. - **Factorise fully** means take out the greatest common factor. - Check by expanding. ### Key Points - Always simplify by collecting like terms after expanding. - In factorising, ensure no common factor remains inside brackets.

Key terms

Expand: Multiply each term inside a bracket by the term outside, removing the brackets.
Factorise: Write an expression as a product of its factors, the reverse of expanding brackets.
Highest Common Factor (HCF): The largest factor that divides all terms; used when factorising.
FOIL: A method for expanding double brackets: multiply First, Outer, Inner, Last terms.
Like terms: Terms with the same variable(s) and power(s); can be combined by adding/subtracting coefficients.

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: − × − = +, − × + = −.

Expand $5(x - 3)$ .

$5x - 3$
$5x - 15$
$5x + 15$
$5x - 8$

Expanding and Simplifying Multiple Brackets

When adding or subtracting brackets, expand each bracket first.
Then collect like terms (terms with the same variable and power).
Example: $4(x+7) + 5x(3-x) = 4x+28 + 15x-5x^{2} = 19x+28-5x^{2}$ .

Expand and simplify: $2(x+5) + 3x(x-8) = ____ x + ____ + ____ x^{2}$

Expanding Double Brackets

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

Expand (x+3)(x+5).

$x^{2}+8x+15$
$x^{2}+15x+8$
$x^{2}+8x+8$
$x^{2}+15x+15$

Expanding a Squared Bracket

Write $(x+3)^{2}$ as (x+3)(x+3) and expand using FOIL.
Example: $(x+3)^{2} = x^{2} + 3x + 3x + 9 = x^{2} + 6x + 9$ .
Do NOT write $(x+y)^{2} =$ x²+y²; it is incorrect.

Which of these are true?

$(x+4)^{2} = x^{2}+8x+16$
$(x-2)^{2} = x^{2}-4x+4$
$(x+3)^{2} = x^{2}+9$

Factorising – Taking Out a Common Factor

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

Factorise $3x + 12$ .

3(x+4)
3(x+12)
x(3+12)
3x(1+4)

Factorising with Variables

Find the HCF of the numbers and the lowest power of each variable.
Example: $5x^{2}y + 25x^{3}y^{2} = 5x^{2}y (x + 5y)$ .
Always check by expanding your answer.

Factorise fully: $30x^{2} - 24x = ____ x (____ x - ____)$

Factorising Completely

Factorise fully means take out the highest common factor.
For example, $6x+10 = 2(3x+5)$ , not x(6x+10) or 2x(3x+5) if x not common.
Check your answer by expanding.

Factorise completely: $8g^{2} - 4g$ .

4g(2g-1)
2g(4g-2)
4g(2g-4)
g(8g-4)

Expanding and Simplifying with Negatives

When expanding, put negative terms in brackets to avoid sign errors.
Example: $(2x-3)(x+4) = 2x^{2} + 8x - 3x - 12 = 2x^{2} + 5x - 12$ .
Collect like terms after expanding.

Expand and simplify: $(x-5)(x-7) = x^{2} - ____ x + ____$

Factorising with Multiple Variables

For expressions like $2a^{3}b^{2} - 4a^{2}b^{3}$ , find HCF of numbers and each variable.
HCF of 2 and 4 is 2; of $a^{3}$ and $a^{2}$ is a²; of $b^{2}$ and $b^{3}$ is $b^{2}$ .
So $HCF = 2a^{2}b^{2}$ , and factorised form is $2a^{2}b^{2}(a - 2b)$ .

Factorise completely: $10a^{2} - 15ab$ .

5a(2a-3b)
$5(2a^{2}-3ab)$
a(10a-15b)
5ab(2a-3)

Flashcards

Expand: $5(x - 3)$

$5x - 15$

Multiply out: $9(3 - x)$

$27 - 9x$

Factorise: $5p + pt$

$p(5 + t)$

Factorise: $3x + 12$

$3(x + 4)$

Factorise: $12x + 15$

$3(4x + 5)$

Factorise: $5y - 6py$

$y(5 - 6p)$

Factorise completely: $18x - 24$

$6(3x - 4)$

Expand: $x(2y + x)$

$2xy + x^{2}$

Expand and simplify: $(x + 3)(x + 5)$

$x^{2} + 8x + 15$

Expand: $x^{2}(x - 7)$

$x^{3} - 7x^{2}$

Factorise: $y^{2} + y$

$y(y + 1)$

Factorise: $2x^{2} - x$

$x(2x - 1)$

Factorise completely: $8g^{2} - 4g$

$4g(2g - 1)$

Factorise: $6y + 21$

$3(2y + 7)$

Factorise fully: $10a^{2} - 15ab$

$5a(2a - 3b)$

Expand and simplify: $4(2m + 3) - 5(m - 2)$

$8m + 12 - 5m + 10 = 3m + 22$

Factorise completely: $12b + 8b^{2}$

$4b(3 + 2b)$

Factorise completely: $21a^{2} + 28ab$

$7a(3a + 4b)$

Expand and simplify: $(x - 5)(x - 7)$

$x^{2} - 12x + 35$

Multiply out: $x(3 - 5x)$

$3x - 5x^{2}$

Factorise fully: $5x - 20x^{2}$

$5x(1 - 4x)$

Factorise completely: $3x^{2} - 12xy$

$3x(x - 4y)$

Expand and simplify: $(m - 3)(m + 2)$

$m^{2} - m - 6$

Factorise completely: $6d^{2}e - 9e^{2}$

$3e(2d^{2} - 3e)$

Factorise completely: $10x^{2} + 24x$

$2x(5x + 12)$

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