2.3.2 Factorisation

Factorisation is the process of writing an algebraic expression as a product of its factors. These factors can be numbers, variables, or an algebraic expression.

In other words, factorisation is the opposite of expanding brackets.

To factorize two terms

• Find the highest common factor of both terms

• Take out the common factor by writing it outside brackets

• Put the remaining inside the brackets

Worked example:

Factorize \( 5m^2-20p^4 \)

Find the highest common factor of \( 5m^2-20p^4 \)

\( 5 \)

Take out \( 5 \) from \( 5m^2-20p^4 \)

\( 5(m^2-4p^4) \)

\( 5(m^2-4p^4) \)

Factoring by grouping:

To factorize such expression

• If the expression has a common bracket

  • Take out the whole bracket like a common factor

• If not, you may require to form the common bracket yourself

  • Take it out the whole bracket like a common factor

Worked example:

Factorise completely.

\( 4fg+6gh+10fk+15hk \)

The expression does not contain the common bracket

So, we need to form the common bracket

\( 4fg+6gh+10fk+15hk \)

Notice that the first two terms have a common factor \( 2g \)

And the last two terms have a common factor \( 5k \)

\( 2g(2f+3h)+5k(2f+3h) \)

Now \( (2f+3h) \) is a common bracket so take it out

\( (2f+3h)(2g+5k) \)

\( (2f+3h)(2g+5k) \)

Worked example:

Factorize \( 2mn+m^2-6n-3m \)

The expression does not contain the common bracket

So, we need to form the common bracket

\( 2mn+m^2-6n-3m \)

Notice that the first two terms have a common factor \( m \)

And the last two terms have \( 3 \)

\( m(2n+m)-3(2n+m) \)

Now \( (2n+m) \) is a common bracket

\( (m-3)(2n+m) \)

\( (m-3)(2n+m) \)