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Factorising Expressions

Factorising Expressions

Factorising Expressions

Factorising is the opposite of expanding brackets, and is a helpful first step in solving equations.

Factorising is the opposite process of expanding brackets. It can be useful in simplifying or solving equations containing algebra.

At its simplest level, factorising is about identifying common factors in terms and writing them outside a bracket. The terms outside and inside the bracket should multiply to make the original expression.

Remember that a common factor is a number or letter which is a factor of all the terms in the expression. Let's start with an expression:


What is the highest common factor(s) of 4x24x^2 and 8x8x?

Now that we've identified the common factors, let's factorise 4x2+8x4x^2+8x.


The highest common factor is 4x4x

Therefore, we can place this outside of a pair of brackets. 4x()4x()


Put terms in the brackets which multiply to the original expression

The goal of factorising is that when we expand the brackets, we will get back to the original expression.


The first term in the brackets should be xx

We need to multiply 4x4x by xx to get the first term in our expression, 4x24x^2. Therefore, our equation becomes 4x(x).4x(x).


By what do we multiply 4x4x by to get 8x8x


We multiply by 2

Therefore, we can put 2 in the brackets alongside what we have so far: 4x(x+2)4x(x+2)


4x(x+2)4x(x+2) is the factorisation of 4x2+8x4x^2+8x

To check this, we need to multiply out of the brackets. If we get back the original expression, we know that we have factorised correctly.


Expand the brackets for 4x(x+2)4x(x+2)


The factorisation is correct!

Expanding the brackets returns the same expression. Therefore, the factorisation of 4x2+8x4x^2+8x is 4x(x+2)4x(x+2).

What is the factorisation of 6x2+15x6x^2 + 15x?

What is the factorisation of 5y30y35y - 30y^3?

Let's try a harder example.

Simplify 3x2y3+6xy23x^2y^3 + 6xy^2


Firstly, we need to find common factors

Common factors are number and letter combinations which are factors of a set of terms. There are 3 parts to each term here: a number, xx and yy.We need to find common factors between the corresponding parts of 3x2y33x^2y^3and6xy26xy^2.


Find the highest common factor of 33 and 66


Find the highest algebraic factor of x2x^2 and xx

The highest algebraic factor is xx, as both terms can be divided by xx.


Find the highest algebraic factor of y3y^3 and y2y^2

The highest algebraic factor isy2y^2. Both of these yy terms can be divided by y2y^2.


Put the common factors together

Our individidual common factors are 33, xx and y2y^2. Together, these make 3xy23xy^2.


Place the common factors, 3xy23xy^2, outside of brackets

3xy2()3xy^2(). We are going to place terms in the brackets which multiply together with 3xy23xy^2 to make each of our original terms.


Place a term in the brackets to multiply to 3x2y33x^2y^3



Place another term in the brackets to multiply to 6xy26xy^2

3xy2(xy+2)3xy^2(xy + 2)


Expand the brackets to check

3xy2(xy+2)=3x2y3+6xy23xy^2(xy + 2) = 3x^2y^3 + 6xy^2


The simplified expression is 3xy2(xy+2)3xy^2(xy + 2)

Nice! This is the final answer.