02Dfactorising

= Factorising = toc

Factorising is the opposite process to expanding brackets.

When we factorise, we put the expression back into brackets.

Taking out a Common Factor

 * Look for the Highest Common Factor (HCF) {the biggest factor that appears in all the terms }
 * Put the HCF out the front of the brackets.
 * Divide each term by the HCF and put the result inside the brackets

To factorise the expression : ab + ac
 * a is the highest common factor
 * a goes in front of the brackets
 * inside the brackets put the result of (ab ÷ a) and then the result of (ac ÷ a)

So, we get : ab + ac = a(b + c)

Notice that if we expand : a(b + c) we get ab + ac {expanding is the reverse of factorising}


 * Example 1 **

math \textbf{(a)} \quad \text{Factorise : } 6a - 15 math

{HCF is 3, so put 3 in front of the brackets, and divide each term by 3}

math . \qquad 6a - 15 = 3 \, \big( 2a - 5 \big) math

math \textbf{(b)} \quad \text{Factorise : } 20m^2 + 15m math

{HCF is 5m, so put 5m in front of the brackets, and divide each term by 5m}

math . \qquad 20m^2 + 15m = 5m \, \big( 4m + 3 \big) math

math \textbf{(c)} \quad \text{Factorise : } -6x^2 - 3xy math Hint : Take out a negative as the common factor

{HCF is –3x, so put –3x in front of the brackets, and divide each term by –3x}

math . \qquad -6x^2 - 3xy = -3x \, \big( 2x + y \big) math

Taking out a bracket
Sometimes the common factor may be a bracket. {the bracket is called a **binomial** because it has 2 terms} The bracket that is common goes out the front and what is left goes in a second bracket.

** Example 2 **
math \textbf{(a)} \quad \text{Factorise : } 4a \, \big( 2x + y \big) + 3 \, \big( 2x + y \big) math

{HCF is (2x + y) so put (2x + y) out the front, and put what is left in the second brackets}



math \textbf{(b)} \quad \text{Factorise : } 5m \, \big( 3m - 1 \big) + \big( 3m - 1 \big) math Hint : In this example, there is an invisible "1" in front of the second bracket.

{HCF is (3m – 1) so put (3m – 1) out the front and put what is left in the second brackets}



Grouping Terms
If the expression has four terms and there are no common factors, it can sometimes be factorised by grouping the terms into pairs and factorising each pair separately.

You may need to rearrange the terms so that pairs with a common factor are together. {don't forget: any minus sign belongs with the term that comes after it}

** Example 3 **
math \textbf{(a)} \quad \text{Factorise : } xy + 5x + 5y + 25 math



math \textbf{(b)} \quad \text{Factorise : } ax - 3y + 3x - ay math



The next step in this sequence is factorising quadratics (Chapter 7)

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