4B - Factorising quadratic trinomials

Methods of factorisation

Factorising using common factors

It is always a good idea to inspect each term for a common factor such as a constant, k, or a simple linear term, ax.

• If all terms have the same common factor it can be removed and the expression factorised.
• Remember: you can always check if the factorised form is correct by expanding!

4B - VIDEO EXAMPLE 1:

Factorise the following expressions:

Factorising using perfect squares

Factorising perfect squares is relatively straight forward; however, you need to be able to recognise them. Recall:

You must recognise:

• The first and last terms appear to be squares (i.e. x² or 81)
• The middle value is twice the value of the two outside terms square root.

4B - VIDEO EXAMPLE 2:

Factorise the following expression:

4B - VIDEO EXAMPLE 3:

Factorise the following expression:

Factorising using the difference of perfect squares (DOPS)

Factorising using the difference of perfect squares (DOPS) is relatively straight forward; however, you need to be able to recognise them. Recall:

You must recognise:

• Whenever there are two square terms (i.e. x² or 81 or 144) with one term subtracted from the other and no middle term present, it can be factorised using DOPS.

4B - VIDEO EXAMPLE 4:

Factorise the following expression:

4B - VIDEO EXAMPLE 5:

Factorise the following expression:

Factorising quadratic trinomials using inspection

A quadratic trinomial has the general form:

Ultimately, when factorised it will have the form:

• The product of m and n gives the value of c: m x n = c
• The sum of of m and n gives the value of b: m + n = b

4B - VIDEO EXAMPLE 6:

Factorise the following expression:

4B - VIDEO EXAMPLE 7:

Factorise the following expression:

4B - VIDEO EXAMPLE 8:

Factorise the following expression:

4B - VIDEO EXAMPLE 9:

Factorise the following expression:

Factorising by grouping

When four or more terms are present, we need to factorise by grouping terms together.

• Which terms to group is done by inspection or trial and error.
• You are aiming to remove a common term from each group and leave the same factor behind (see examples below).
• Often when grouping we need to group so that other methods of factorisation can be used, such as:
  • Difference of perfect squares (DOPS)
  • Perfect squares
  • Inspection

Grouping 2 and 2

4B - VIDEO EXAMPLE 10:

Factorise the following expression:

4B - VIDEO EXAMPLE 11:

Factorise the following expression:

Grouping 3 and 1

4B - VIDEO EXAMPLE 12:

Factorise the following expression:

Factorising using CAS

CAS calculators can readily factorise polynomials using the following commands:

• Factor - use when the factors are rational.
• rFactor - used when the factors are irrations.

Try to factorise using Factor first, if this does not work use rFactor.

Factor

The factor command is located under transformations.

Figure 1 - Location of the factor command

The factor command is used to show the factors of an expression. It is essentially the opposite to expand.

Figure 2 - The factor command in action.

rFactor

The rFactor command is located under transformations.

Figure 3 - Location of the rFactor command

The rFactor command stands for "root factor" and is used to show the factors of an expression when a surd is present. For quadratics this occurs when the discriminant (Δ) is not a perfect square.

Important: If you attempt to factorise and expression with irrational roots (surds) using the factor command, instead of rFactor, it will return the same expression (unfactorised) - see example below.

Note: Do not attempt to use rFactor over the complex set of number.

Figure 4 - The factor and rFactor command in action on the same expression.