How to factorise using brackets

The factorisation process is the opposite of expanding the bracket. An algebraic expression for factorising means put the expression into the brackets by taking the common factors. For example, $$4x+12=4(x+3)$$, where $$4x+12$$ is the expanding brackets, and $$4(x+3)$$ is the factorisation.. The simplest way of factorising is: 

1. Find the greatest common factor of each term in the expression.

2. Write HCF, i.e., the highest common factor in front of the bracket.

3. Multiply each item in the parentheses.

A real-life example of factorising is calculating time. Every day contains $$ 24\text{ hours}$$, and you must take vitamin tablets $$3$$ times a day. Then, you will be calculating time with the help of factorisation, as $$3 \times 8=24$$. So, you will take $$1$$ tablet after every $$8\text{ hours}$$ in a day.

There are three methods to factorise expression. They are factorising single brackets, factorising double brackets and differences of two squares. They all are discussed below.

E2.2: Use brackets and extract common factors. 

Methods to factorise expression

There are three methods to factorise expression. They are discussed below:

Factorising single brackets

In factorising single brackets, the expression will be in the form $$a(bx+c)$$, where $$a$$, $$b$$ and $$c$$ are the three constants. $$a(bx+c)$$ is known as binomial, as it is a two-term expression.

For example, $$5x+15=5(x+3)$$, where $$5x+15$$ is the expanding bracket, and $$5(x+3)$$ is the factorising single brackets.

Factorising double brackets 

Factorising double brackets means factorising a quadratic equation in the form $$x^{2}+bx+c$$ or $$ax^{2}+bx+c$$. These are known as trinomials.

 The factorisation of $$x^{2}+3x+2=(x+1)(x+2)$$ is an example of $$x^{2}+bx+c$$ and $$2x^{2}+5x+2=(2x+1)(x+2)$$ is an example of $$ax^{2}+bx+c$$. Here, $$x^{2}+3x+2$$ and $$2x^{2}+5x+2$$ are the expanding brackets; $$(2x+1)(x+2)$$ and $$(x+1)(x+2)$$ are the factors.

Differences of two squares

The difference of two squares is a method of factorising that is used when an algebraic expression consist of two squared terms and one is subtracted from the other term. It is represented as $$a^{2}-b^{2}=(a+b)(a-b)$$.

For example, $$x^{2}-9=(x+3)(x-3)$$, where $$x^{2}-9$$ is the expanding brackets and $$(x+3)(x-3)$$ is the factorising.

A real-life example of factorisation is travelling. Suppose you want to travel to a historic place. You know the distance between you and that historical place is $$480\text{ miles}$$. You want to know how many hours you will take to reach there if you take your car at an average speed of $$60\text{ mph}$$. It will take $$8\text{ hours}$$ to reach there. This is calculated with the help of factorisation, as $$60\text{ mph} \times 8\text{ hours}=480\text{ miles}$$.

Worked examples of factorising algebraic expressions

Example 1: Factorise the expression $$15-3y$$.

Step 1: Given information

$$15-3y$$

Step 2: Find the highest common factor.

The highest common factor is $$3$$ between $$15$$ and $$3$$. 

Step 3: Take $$3$$ as a common factor in the whole expression.

$$15-3y=3(5-y)$$

So, $$3(5-y)$$ is the factor of the expression $$15-3y$$.

Example 2: Factorise the expression $$10x-5x^{2}$$.

Step 1: State the given information

$$10x-5x^{2}$$

Step 2: Find the highest common factor.

The highest common factor is $$5x$$ between $$10x$$ and $$5x^{2}$$. 

Step 3: Take $$5x$$ as a common factor in the whole expression.

$$10x-5x{2}=5x(2-x)$$

So, $$5x(2-x)$$ is the factor of the expression $$10x-5x{2}$$.

Example 3: Factorise the expression $$x^{2}-625$$.

Step 1: State the given information

$$x^{2}-625$$

Step 2: Use difference of two square.

Take square root of $$x^{2}$$ and $$625$$.

$$\sqrt{x^{2}}=x$$

$$\sqrt{625}=25$$

Step 3: Write the final factors.

$$x^{2}-625=(x+25)(x-25)$$

So, $$(x+25)(x-25)$$ is the factor of the expression $$x^{2}-625$$.

Example 4: Factorise the expression $$5x^{2}+16x+3$$.

Step 1: Given information

$$5x^{2}+16x+3$$

Step 2: Use factorising double brackets method.

Look for the two numbers whose product is $$15$$ and sum is $$16$$.

The numbers are $$15$$ and $$1$$.

Step 3: Write the final factor.

$$5x^{2}+16x+3=(5x+1)(x+3)$$

So, $$(5x+1)(x+3)$$ is the factor of the expression $$5x^{2}+16x+3$$.