Algebra

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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:

$4x^2+8x$

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

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

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, $4x^2$ . 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 $4x^2+8x$

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 $4x(x+2)$

The factorisation is correct!

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

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

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

Let's try a harder example.

Simplify $3x^2y^3 + 6xy^2$

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, $x$ and $y$ .We need to find common factors between the corresponding parts of $3x^2y^3$ and $6xy^2$ .

Find the highest common factor of $3$ and $6$

Find the highest algebraic factor of $x^2$ and $x$

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

Find the highest algebraic factor of $y^3$ and $y^2$

The highest algebraic factor is $y^2$ . Both of these $y$ terms can be divided by $y^2$ .

Put the common factors together

Our individidual common factors are $3$ , $x$ and $y^2$ . Together, these make $3xy^2$ .

Place the common factors, $3xy^2$ , outside of brackets

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

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

$3xy^2(xy)$

Place another term in the brackets to multiply to $6xy^2$

$3xy^2(xy + 2)$

Expand the brackets to check

$3xy^2(xy + 2) = 3x^2y^3 + 6xy^2$

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

Nice! This is the final answer.