YOU ARE LEARNING:
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 4x2 and 8x?
Now that we've identified the common factors, let's factorise 4x2+8x.
The highest common factor is 4x
Therefore, we can place this outside of a pair of brackets. 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 x
We need to multiply 4x by x to get the first term in our expression, 4x2. Therefore, our equation becomes 4x(x).
By what do we multiply 4x by to get 8x
We multiply by 2
Therefore, we can put 2 in the brackets alongside what we have so far: 4x(x+2)
4x(x+2) is the factorisation of 4x2+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)
The factorisation is correct!
Expanding the brackets returns the same expression. Therefore, the factorisation of 4x2+8x is 4x(x+2).
What is the factorisation of 6x2+15x?
What is the factorisation of 5y−30y3?
Let's try a harder example.
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, x and y.We need to find common factors between the corresponding parts of 3x2y3and6xy2.
Find the highest common factor of 3 and 6
Find the highest algebraic factor of x2 and x
The highest algebraic factor is x, as both terms can be divided by x.
Find the highest algebraic factor of y3 and y2
The highest algebraic factor isy2. Both of these y terms can be divided by y2.
Put the common factors together
Our individidual common factors are 3, x and y2. Together, these make 3xy2.
Place the common factors, 3xy2, outside of brackets
3xy2(). We are going to place terms in the brackets which multiply together with 3xy2 to make each of our original terms.
Place a term in the brackets to multiply to 3x2y3
Place another term in the brackets to multiply to 6xy2
Expand the brackets to check
The simplified expression is 3xy2(xy+2)
Nice! This is the final answer.