Algebra: A Step-By-Step Guide To Factorising Linear Expressions

Introduction

Are you wondering why you are not awarded any marks when asked to factorise an algebraic linear equation despite following the steps you learned in school?

You might be doing an important step wrong!

In this blog post, I will be sharing a step-by-step guide to factorising linear expressions to help you get full marks in your Algebra questions.

How Are Algebraic Expansion And Algebraic Factorisation Related?

In the previous blog post, we talked about expanding linear expressions.

If you haven’t read that, click the link and go over it first because algebraic expansion and algebraic factorisation are related.

To give you a brief recap, this is what happens when you expand linear expressions.

Meanwhile, when you are asked to factorise an algebraic expression, you are supposed to go in the opposite direction.

This time, from 2x + 6, you are supposed to take the highest common factor (HCF) and put it outside the bracket so you will get 2(x + 3).

If this looks confusing, keep on reading because I will explain this in detail.

Let’s Take A Look At This Algebra Question

Let’s Solve Part (A)

The first step in factorising an algebraic expression is to find the highest common factor (HCF).

🤔 What Is HCF? 🤔

HCF is the largest number that can divide given numbers without leaving a remainder.

Some of you might think that 5 is the HCF because it can divide 15 and 30. So let us try that.

Let us put 5 outside the bracket. What will give you 15x?

5(3x)

Next, with 5 outside the bracket, what will give you 30?

5(6)

Now, let us take all the numbers inside the bracket.

=5(3x + 6)

If this is your answer, then I can guarantee that you will get zero marks because this is incorrect!

The correct way to do it is to copy the question and write down the HCF.

15x + 30

What is the HCF of 15 and 30? It is 15!

So let us put 15 outside the bracket. What will give you 15x?

15(x)

Next, with 15 outside the bracket, what will give you 30?

15(2)

Now, let us take all the numbers inside the bracket.

=15(x + 2)

Therefore, the correct answer is 15(x +2).

Answer For Part (A)

15(x +2)

Let’s Solve Part (B)

Some of you might be panicking because you don’t see any numbers in this algebraic equation. Don’t worry because I am going to show you how to tackle this question!

What do p² and pq² mean?

Let’s recall, how do you write 11 x 11 in a short form?

You will write 11². Why? Because there are two 11s.

Similarly, if you have p x p, it will become p² . If it is a q x q, it will become q².

So, what is the HCF of  p² + pq²?

It is p! Because after you have taken out p, there is nothing else that p and q² have in common.

Now, let us put p outside the bracket. What will give you p²?

p(p)

Next, put p outside the bracket, what will give you pq²?

p(q²)

Now, let us take all the numbers inside the bracket.

= p(p + q²)

Answer For Part (B)

p(p + q²)

Let’s Solve Part (C)

First, we have to find the HCF of:

• 21 and 14
• x³ and x²

The HCF of 21 and 14 is 7!

What about x³ and x²?

x³ is x • x • x

x² is x • x

So, their HCF is x².

Therefore, our overall HCF will be 7x².

Now, let us put 7x² outside the bracket. What will give you -21x³?

7x²(-3x)

Next, put 7x². What will give you 14x²?

7x²(2)

Now, let us take all the numbers inside the bracket.

= 7x²(-3x+2)

However, there is another way to answer this question. Instead of pulling out 7x², you can pull out -7x².

Let us now apply the same steps. Let us put -7x² outside the bracket. What will give you -21x³?

-7x²(3x)

Next, put -7x² outside the bracket, what will give you 14x²?

-7x²(-2)

Now, let us take all the numbers inside the bracket.

=-7x(3x-2)

Answer For Part (C)

7x²(-3x+2) or -7x(3x-2)

Let’s Solve Part (D)

You will notice that all the terms here are negative.

💡 Remember This When All Your Algebraic Terms Are Negative💡

When your algebraic terms are negative, you have to extract a negative factor.

What do -16, -12, and -4 have in common?

They are all factors of -4. So our HCF is -4.

Let us put -4 outside the bracket. What will give you -16x?

-4(4x)

Next, put -4 outside the bracket. What will give you -12y?

-4(3y)

Lastly, if I already pulled out a -4, how will I get back -4?

-4(1)

Now, let us take all the numbers inside the bracket.

= -4(4x +3y + 1)

Answer For Part (D)

-4(4x +3y + 1)

Conclusion

I hope that after reading this Algebra blog post, you have gained an understanding of how to correctly factorise linear expressions.

Keep in mind to be careful when identifying the Highest Common Factor (HCF) to avoid losing marks. Remember that HCF is the largest number that can divide given numbers without leaving a remainder

Stay tuned for more blog posts that help you tackle challenging Algebra problems.