Factorising Revision Notes for Grade 12 NSC Matric Mathematics

Factorising

What is factorising?

Factorising is the process of rewriting an algebraic expression as a product of its factors. This fundamental algebraic technique transforms expressions from their expanded form into a form where terms are multiplied together. Factorising is the reverse operation of expanding brackets and serves as a powerful tool for simplifying complex expressions and solving equations efficiently.

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Factorising is one of the most important skills in algebra. It's the key to solving quadratic equations, simplifying complex fractions, and understanding polynomial behaviour. Master this skill early, as it forms the foundation for advanced mathematics.

The primary purpose of factorising is to make algebraic manipulation easier. When expressions are factorised, they become much simpler to work with, especially when solving equations, finding roots, or performing further algebraic operations.

Methods of factorising

Taking out the common factor

The first and most important step in any factorising problem is to identify and extract the highest common factor (HCF). This method involves finding the largest factor that divides into all terms of the expression and factoring it out from each term.

To apply this method effectively:

Examine all terms in the expression
Identify the highest power of each variable that appears in every term
Find the largest numerical factor that divides all coefficients
Factor out this HCF from the entire expression

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Worked Example: Taking Out Common Factors

Let's factorise: $9x^2 - 6xy^2$

Step 1: Identify the HCF of the coefficients: HCF of 9 and 6 is 3

Step 2: Identify the HCF of the variables: $x$ appears in both terms, with the lowest power being $x^1$

Step 3: Factor out $3x$ : $9x^2 - 6xy^2 = 3x(3x - 2y^2)$

Factorising by grouping in pairs

This method is particularly useful for expressions with four terms. The technique involves grouping terms strategically so that each group has a common factor, then factoring out these common factors to reveal a final common bracketed expression.

The process follows these steps:

Group the terms into pairs
Factor out the common factor from each pair
Look for a common bracketed expression
Factor out this common expression completely

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Worked Example: Grouping in Pairs

Let's factorise: $3xy - 2x + 3y - 2$

Step 1: Group terms in pairs: $(3xy - 2x) + (3y - 2)$

Step 2: Factor out common factors from each group: $x(3y - 2) + 1(3y - 2)$

Step 3: Factor out the common bracketed expression: $(x + 1)(3y - 2)$

Factorising the difference of two squares

The difference of two squares is a special algebraic pattern that appears frequently in mathematics. This method applies when you have two perfect squares separated by a minus sign.

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Key Formula: $a^2 - b^2 = (a - b)(a + b)$

This formula works because when you expand $(a - b)(a + b)$ , the middle terms cancel out, leaving only $a^2 - b^2$ .

To identify this pattern:

Look for two terms separated by subtraction
Check that both terms are perfect squares
Apply the formula directly

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Worked Example: Difference of Two Squares

Let's factorise: $16x^2 - y^2$

Step 1: Identify the perfect squares: $16x^2 = (4x)^2$ and $y^2 = (y)^2$

Step 2: Apply the formula: $16x^2 - y^2 = (4x - y)(4x + y)$

Factorising the difference of two cubes

The difference of two cubes follows a specific pattern that's essential to memorise. This method applies when you have two perfect cubes separated by subtraction.

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Key Formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

The second bracket follows a specific pattern: it starts with $a^2$ , then has $+ab$ , and ends with $+b^2$ . Notice that the signs in the second bracket are both positive.

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Worked Example: Difference of Two Cubes

Let's factorise: $8x^3 - y^3$

Step 1: Identify the perfect cubes: $8x^3 = (2x)^3$ and $y^3 = (y)^3$

Step 2: Apply the formula: $8x^3 - y^3 = (2x - y)(4x^2 + 2xy + y^2)$

Factorising the sum of two cubes

The sum of two cubes has a similar but distinct pattern from the difference of cubes. This method applies when you have two perfect cubes separated by addition.

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Key Formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

The key difference from the difference of cubes is in the signs: the first bracket has addition, while the second bracket alternates signs starting with positive.

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Worked Example: Sum of Two Cubes

Let's factorise: $27a^3 + 64b^3$

Step 1: Identify the perfect cubes: $27a^3 = (3a)^3$ and $64b^3 = (4b)^3$

Step 2: Apply the formula: $27a^3 + 64b^3 = (3a + 4b)(9a^2 - 12ab + 16b^2)$

Factorising trinomials

Trinomials are expressions with three terms, commonly appearing in the form $ax^2 + bx + c$ . The method for factorising trinomials involves finding two numbers that satisfy specific conditions related to the coefficients.

For trinomials of the form $ax^2 + bx + c$ :

Find two numbers that multiply to give $ac$ (the product of the first and last coefficients)
These same two numbers must add to give $b$ (the middle coefficient)
Use these numbers to split the middle term and then group

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When factorising trinomials, patience is key. Sometimes it takes several attempts to find the right pair of numbers. Don't get discouraged if your first attempt doesn't work!

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Worked Example: Factorising Trinomials

Let's factorise: $9x^2 + 5x - 4$

Step 1: Identify $a = 9$ , $b = 5$ , $c = -4$

Step 2: Find two numbers that multiply to $(9)(-4) = -36$ and add to $5$

Step 3: These numbers are $9$ and $-4$ (since $9 \times (-4) = -36$ and $9 + (-4) = 5$ )

Step 4: The factorised form is: $(9x - 4)(x + 1)$

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Summary of Factorising Techniques

When approaching any factorising problem, follow this systematic approach to ensure success:

Always factor out the highest common factor (HCF) first - this is the most crucial first step that can simplify the entire process
Look for special patterns such as the difference of squares, sum of cubes, or difference of cubes - recognising these patterns allows for immediate application of standard formulas
Use grouping for four-term expressions - when you have exactly four terms, grouping in pairs often reveals common factors
For trinomials, find two numbers that multiply to the constant term and add to the middle coefficient - this systematic approach works for most quadratic expressions

These techniques form the foundation of algebraic manipulation and are essential tools for solving equations, simplifying expressions, and preparing for more advanced mathematical concepts.

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Remember - Essential Tips for Success

Factor first - always look for common factors before attempting other methods
Recognise patterns - difference of squares and cubes have specific formulas to memorise
Use systematic approaches - for trinomials, focus on finding the two key numbers
Check your work - expand your factorised answer to verify it matches the original expression
Practice identifying which method to use by examining the number and type of terms in the expression

Factorising (Grade 12 NSC Matric Mathematics): Revision Notes

Factorising

What is factorising?

Methods of factorising

Taking out the common factor

Factorising by grouping in pairs

Factorising the difference of two squares

Factorising the difference of two cubes

Factorising the sum of two cubes

Factorising trinomials

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