Algebraic Expansion and Factorisation Revision Notes for HSC SSCE Mathematics Advanced

Algebraic Expansion and Factorisation

Algebraic expansion and factorisation are fundamental skills in Year 11 Mathematics that you'll use throughout the course. Understanding these inverse processes will help you manipulate and simplify algebraic expressions with confidence.

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Key idea

Expansion means multiplying out brackets to remove them from an expression. When you expand, you distribute terms across the bracket.

Factorisation is the reverse process. It involves rewriting an expression by putting it back into brackets.

These are inverse processes, meaning they undo each other. You can always check your factorisation by expanding it again.

Expanding expressions

Expansion is about removing brackets from algebraic expressions. There are several techniques depending on the type of expression you're working with.

Expanding a single bracket

When expanding a single bracket, distribute the term outside the bracket to each term inside. This follows the distributive law:

$a(b + c) = ab + ac$

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Worked Example: Single Bracket Expansion

Expand $3(x + 4)$

Solution:

Multiply $3$ by each term inside the bracket:

$= 3 \times x + 3 \times 4$

$= 3x + 12$

Expanding two brackets

When expanding two brackets, multiply each term in the first bracket by each term in the second bracket. A systematic approach ensures you don't miss any terms.

Consider the expansion of $(x + 2)(x + 3)$ :

Distribute each term from the first bracket:

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

Now expand each part:

$= x^2 + 3x + 2x + 6$

Combine like terms:

$= x^2 + 5x + 6$

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Tip: Make sure every term in the first bracket multiplies every term in the second bracket. This systematic approach prevents missing terms.

Expanding with negatives

When negative signs are involved, you must pay careful attention to signs throughout the calculation.

Consider $(x - 4)(x + 2)$ :

$= x(x + 2) - 4(x + 2)$

$= x^2 + 2x - 4x - 8$

$= x^2 - 2x - 8$

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Common error: Forgetting that $-4 \times 2 = -8$ , not $+8$ . Always track negative signs carefully at each step.

Special case: Perfect squares

Perfect squares follow specific patterns that are useful to recognise and memorise:

$(x + a)^2 = x^2 + 2ax + a^2$

$(x - a)^2 = x^2 - 2ax + a^2$

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Worked Example: Perfect Square Expansion

Expand $(x + 5)^2$

Solution:

Using the perfect square pattern:

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

$= x^2 + 10x + 25$

Factorising expressions

Factorisation reverses the expansion process by putting expressions into brackets. This skill is essential for solving equations and simplifying expressions.

Factorising by taking out a common factor

Identify the highest common factor (HCF) that divides all terms in the expression, then take it outside the bracket.

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Worked Example: Common Factor

Factorise $6x + 12$

Solution:

Both terms are divisible by $6$ :

$= 6(x + 2)$

Factorising quadratics

For a quadratic in the form $x^2 + bx + c$ , you need to find two numbers that:

Multiply to give $c$
Add to give $b$

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Worked Example: Simple Quadratic Factorisation

Factorise $x^2 + 5x + 6$

Solution:

Find two numbers that multiply to $6$ and add to $5$ :

The numbers are 2 and 3 because $2 \times 3 = 6$ and $2 + 3 = 5$

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

Quadratics with coefficient of $x^2 \neq 1$

When the coefficient of $x^2$ is not $1$ , use the splitting method.

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Worked Example: Advanced Quadratic Factorisation

Factorise $2x^2 + 7x + 3$

Solution:

Step 1: Multiply the coefficient of $x^2$ by the constant term

$2 \times 3 = 6$

Step 2: Find two numbers that multiply to $6$ and add to $7$

The numbers are 6 and 1

Step 3: Split the middle term using these numbers

$2x^2 + 6x + x + 3$

Step 4: Factorise in pairs

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

Step 5: Take out the common bracket

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

Difference of two squares

This special pattern appears when you have two perfect squares separated by a minus sign:

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

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Worked Example: Difference of Two Squares

Factorise $x^2 - 16$

Solution:

Recognise that $x^2$ and $16$ are both perfect squares:

$= (x - 4)(x + 4)$

Worked examples

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Worked Example 1: Expand Fully

Expand $(2x + 3)(x + 5)$

Solution:

Step 1: Multiply each term in the first bracket by each term in the second bracket

$= 2x \cdot x + 2x \cdot 5 + 3 \cdot x + 3 \cdot 5$

$= 2x^2 + 10x + 3x + 15$

Step 2: Simplify by combining like terms

$= 2x^2 + 13x + 15$

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Worked Example 2: Factorise a Quadratic

Factorise $x^2 - x - 6$

Solution:

Step 1: Find two numbers that multiply to $-6$ and add to $-1$

The numbers are -3 and 2 because:

$-3 \times 2 = -6$
$-3 + 2 = -1$

Answer:

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

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Worked Example 3: Factorise (Harder)

Factorise $3x^2 - 11x - 4$

Solution:

Step 1: Multiply $a \times c$

$3 \times (-4) = -12$

Step 2: Find two numbers that multiply to $-12$ and add to $-11$

The numbers are -12 and 1

Step 3: Split the middle term

$3x^2 - 12x + x - 4$

Step 4: Factorise in groups

$= 3x(x - 4) + 1(x - 4)$

Step 5: Take out the common factor

$= (3x + 1)(x - 4)$

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Common mistakes

Missing terms when expanding

Always multiply every term in the first bracket by every term in the second bracket. A systematic approach prevents missing terms.

Sign errors

Be especially careful with negative signs. When multiplying or distributing negatives, track signs at each step.

Incorrect factorisation

Always verify your factorisation by expanding the brackets again. If you get back to the original expression, your factorisation is correct.

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Exam tips

Use a clear structure when expanding

Write out each multiplication step rather than doing multiple steps in your head. This reduces errors and makes your working easier to follow.

For quadratics

Write out the factor pairs of $c$ (or $a \times c$ for harder quadratics) to systematically find the correct pair.

Always check your answer

Expand your factorised brackets to confirm you get back to the original expression.

Watch for special patterns

Perfect squares and difference of two squares save time in exams when you recognise them quickly.

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Remember!

Key Points to Remember:

Expansion means multiplying out brackets to remove them, while factorisation is the reverse process of putting expressions into brackets.
When expanding two brackets, multiply every term in the first bracket by every term in the second bracket systematically.
For simple quadratics ( $x^2 + bx + c$ ), find two numbers that multiply to $c$ and add to $b$ .
For harder quadratics where the coefficient of $x^2 \neq 1$ , use the splitting method: multiply $a \times c$ , find factor pairs, split the middle term, and factorise in groups.
Recognise special patterns like perfect squares $(x \pm a)^2 = x^2 \pm 2ax + a^2$ and difference of two squares $a^2 - b^2 = (a - b)(a + b)$ to work more efficiently in exams.

Algebraic Expansion and Factorisation (HSC SSCE Mathematics Advanced): Revision Notes