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In Algebra, factorising is the opposite of expanding.

Prior Knowledge – M1 Factorising

Factorising expressions

We know that we can expand the expression \(2p (q + 4)\) to give \(2pq + 8p\)

Expanding and Factorising are inverse operations.
This means that we can factorise \(2pq + 8p\) to give \(2p (q + 4)\)

To factorise an expression

• Look for a common factor and place this outside a bracket
• Work out what needs to go inside the bracket to keep the expression correct when multiplied out
• Keep the sign between the terms the same

Example

Factorise \(9m – 12m²\)

Solution

Look for a common factor and place this outside a bracket

The highest common factor of 9m and 12m² is ‘3m’

Think about the numbers and the ‘m’ terms separately.
'3' is the HCF of 9 and 12
'm' is the HCF of m and m²

3m goes outside the bracket.

\(15m – 12m² = 3m ( ? )\)

Work out what needs to go inside the bracket

Keep the sign between the terms the same

\(15m – 12m² = 3m (5 – 4m)\)

Answer

\(15m – 12m² = 3m (5 – 4m)\)

Check by expanding

\(3(5m – 4m) = 15m – 12m²\)

Question

Factorise \(5tm – 35m\)

Solution:

Look for a common factor and place this outside a bracket

The highest common factor of 5tm and 35m is 5m

\(5tm – 35mn = 5m ( ? )\)

Work out what needs to go inside the bracket

\(5tm – 35m = 5m (t – 7)\)

Answer:

\(5m (t – 7)\)

Check

\(5(t – 7m) = 5t – 35m\)

Example

Factorise this expression fully \(4q – 12q^2\)

Solution

One common factor of 4 and 12 is 2

\(4q – 12q² = 2(2q – 6q²)\)

While this statement is true, the expression has not been fully factorised.
To do this, the common factor outside the bracket must be the highest common factor.
The HCF of \(4\) and \(12\) is \(4\) and the HCF of \(q\) and \(q^2\) is \(q\)

Answer

\(4q – 12q^2 = 4q (1 – 3q)\)

Check

\(4q (1 – 3q) = 4q – 12q^2\)

Question

Factorise fully \(15x - 4xy - 7x^2\)

Solution:

\(x\) is the only factor common to all three terms

\(15x - 4xy – 7x^2 = x (15 - 4y - 7x)\)

Answer:

\(15x - 4xy – 7x^2 = x (15 - 4y - 7x)\)