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In order to access this I need to be confident with:
Expanding Brackets
Factors and Multiples
Terms, Expressions, Equations, Formulas
Adding and Subtracting Negative numbers
Multiplying Negative Numbers
Highest Common Factor (HCF)
Multiplying Algebraic Terms
Laws of Indices
Square Numbers
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Here is everything you need to know about factorising for GCSE maths (Edexcel, AQA and OCR). You’ll learn the essentials of factorising expressions and factorising quadratics including factorising into single brackets and double brackets.
Look out for the factorising worksheets and exam questions at the end
Factorising is the reverse process of expanding brackets. To factorise an algebraic expression means to put it in brackets by taking out the common factors.
The simplest form of factorising is
However you will need to learn various ways of factorising expressions depending on the examples you’re presented with.
To factorise algebraic expressions there are three basic methods. When you are factorising quadratics you will usually use the double brackets or difference of two squares method.
1. Factorising single brackets
Example of factorising an algebraic expression:
3x + 6 = 3(x + 2)
2. Factorising double brackets
a) Example of factorising quadratic expressions in the form
x2 + 6x + 5 = (x + 5)(x + 1)
Remember:
Expressions with three terms like
b) Example of factorising quadratic expressions in the form
2x2 + 5x + 3 = (2x + 3)(x + 1)
3. Factorising differences of two squares
Example of factorising the difference of two squares:
4x2 – 16 = (2x – 4)(2x + 4)
Each method of factorising expressions is summarised below. For detailed examples, practice questions and worksheets on each one follow the links to the step by step guides.
Factorising example using single brackets
To fully factorise:
3x + 6
Factors of 3:
1, 3
Factors of 6:
1, 6
2, 3
Top tip:
Writing the factor pairs makes it easier to list all the factors
The highest common factor (HCF) of
2 Write the highest common factor (HCF) at the front of the single bracket.
3 Fill in each term in the bracket by multiplying out.
What do I need to multiply
What do I need to multiply
We can check the answer by multiplying out the bracket!
Step by step guide: Factorising single brackets
Factorising example for quadratic expressions in the form
To fully factorise:
Factors of 5:
1, 5
2 Find a pair of factors that + to give the middle number (6) and ✕ to give the last number (5).
3 Write two brackets and put the variable at the start of each one.
4 Write one factor in the first bracket and the other factor in the second bracket. The order isn’t important, the signs of the factors are.
Factorising example for quadratic expressions in the form
To fully factorise:
2 We need a pair of factors that + to give the middle number (5) and ✕ to give this new number (6)
3 Rewrite the original expression, this time splitting the middle term into the two factors we found in step 2.
4 Split the equation down the middle and fully factorise each half.
5 Factorise the whole expression by bringing whatever is in the bracket to the front and writing the two other terms in the other bracket.
Step by step guide: Factorising quadratics
Factorising example using difference of two squares:
To fully factorise:
2 Square root the first term and write it on the left-hand side of both brackets.
3 Square root the last term and write it on the right-hand side of both brackets.
4 Put + in the middle of one bracket and – in the middle of the other (the order doesn’t matter).
Step by step guide: Difference of two squares
1. Fully factorise: 10 – 5y
=5(2 – y)
2. Fully factorise: 20x2 – 8x
=4x(5x -2)
3. Fully factorise: x2 – x – 6
=(x + 2)(x – 3)
4. Fully factorise: 2x2 – 4x – 6
=(2x + 2)(x – 3)
5. Fully factorise: x2 – 9
=(x + 3)(x-3)
6. Fully factorise: 4x2 – 16
=(2x + 4)(2x – 4)
1. Factorise 9x – 18
9(x – 2)
(1 mark)
2. Factorise fully 16x2 + 20xy
4x(4x + 5y)
(2 marks)
3. Factorise fully 3y2 – 4y – 4
(3y + 2)(y – 2)
(2 marks)
Get your free factorising worksheet of 20+ questions and answers. Includes reasoning and applied questions.
You have now learnt how to:
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