Algebra

Factorisation I

Factorisation is an essential tool when you start to meet difficult equations to solve, like quadratic. The rule of factorisation is that:

ax + bx = (a + b)x

This rule may seem quite tedious, but when you get the hang of it, it becomes quite simple.

Let's view an example:

Question 8: Evaluate 2x + 3x.

How can we relate these two expressions?

2x + 3x

AND

ax + bx

Well, we could say a = 2, and b = 3 to get:

ax + bx = 2x + 3x

And this leads us to the rule:

ax + bx = (a + b)x

2x + 3x = (2 + 3)x

2x + 3x = 5x

Let's do more examples:

Question 9: Evaluate 3x + 2x

Answer: 5x

Question 10: Evaluate 3x + 6x

Answer: 9x

Question 11: Evaluate x + 2x 

Answer: 

3x

Let's bend the rule a bit so that we can evaluate more expressions.

Here is the original rule:

ax + bx = (a + b)x

Let's define a new variable c such that:

c = - b

We could multiply both sides by -1 to get:

-1 × c = -1 × - b

- c = b

b = -c

So that means we could replace b with - c to get:

ax + bx = (a + b)x

ax + (- c)x = (a + (- c))x

ax - cx = (a - c)x

So this is our new rule:

ax - cx = (a - c)x

Let's do a couple of example questions

Question 12: Evaluate 3x - 2x

Answer: Using the rule:

ax - cx = (a - c)x

We could substitute a = 3 and c = 2 to get:

3x - 2x = (3 - 2)x

3x - 2x = (1)x

3x - 2x = x

So the answer is x.

Before you move on to Factorisation II, I'd highly recommend you to be able to this factorisation in your head.

Question 13: 2x + 1 = 3x - 2. Solve for x

Answer: This question may be a bit hard, so I will solve it for you.

2x + 1 = 3x - 2

2x +1 + 2 = 3x -2 + 2

2x + 3 = 3x

2x + 3 - 2x = 3x - 2x

2x - 2x + 3 = (3 - 2)x

0 + 3 = x

3 = x

So x is equal to 3.