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Algebra (Intermediate) - Factorising
 

In algebra, we factorise by putting brackets into the expression.

There are three methods of factorising. 

Method 1: Common Factor

Example 1 : Factorise 3y + 6

In this case, 3 is a common factor because it will divide into 3y and 6

Therefore the answer is written as

In the next example there are two common factors.

Example 2: Factorise 5y² - 10y

5 and y are factors, so the answer is

When asked to factorise, we must take out all the common factors.
 

Method 2: Factorising Quadratic Expressions Containing Three Terms

A quadratic expression contains a squared term as its highest power.

Example 1: Factorise y² + 5y + 6

In this method we use two brackets. Inside these we put the factors of the first and last terms as follows:

Note: there are two pairs of factors of the last term in this case. We choose the pair which add up to the middle term (in this case, 5).

So the answer is (y +3 )(y + 2)

The signs of the terms can be different, as in the following case.

In this case, only 5 + -2 = 3 (which is the middle term)

Answer = (y +5 )( y -2)

Method 3: Difference Between Two Squares

This method is for expressions containing two squared terms and the sign between them must be minus.

Example 1: Factorise x²  - 9

Both x squared and 9 are squared terms and a minus is between them so we can factorise in this way.

Note: we can check any answer by removing the brackets. This is called 'expanding brackets'.

For example, 3(y + 2) = 3 x y + 3 x 2 = 3y + 6

Note: we multiply everything inside the bracket by 3.

Double Brackets

Note: we multiply the second bracket by y and then by 3.