Lucan Community College Leaving Certificate Mathematics Higher Level

Lucan Community College Leaving Certificate Mathematics Higher Level Mr Duffy & Ms Mc. Kelvey The Factor Theorem © Ciarán Duffy

The Factor Theorem A simple quadratic function can be factorised by inspection e. g. Consider The factors are From the factors, we can see that the zeros of the functions are and So, Reversing this process enables us to factorise cubics ( and polynomials of higher degrees )

The Factor Theorem e. g. If is a factor ( Notice the change of sign ) For a polynomial function says that: if then , the factor theorem is a factor

The Factor Theorem Factorising a Cubic Function e. g. 1 Find a linear factor of Solution: Let Try a = 1: is a factor Once we have found one factor of a cubic, the other factor, which is quadratic, can be found by inspection. Sometimes this quadratic factor will also factorise

The Factor Theorem e. g. 2 Factorise fully Solution: We use the factor theorem to find one linear factor: Let The linear factor can only contain numbers that are factors of 8. We could try any of is not a factor These numbers suggest that So, is a factor could be a factor quadratic factor The quadratic factor can be found by inspection.

The Factor Theorem We first get the and +8

The Factor Theorem We next need but we already have So, we need

The Factor Theorem The linear term can be used to check the result. Finally, we need to check whether the quadratic factor will factorise into 2 linear factors. In this example there are no further factors. We have

The Factor Theorem SUMMARY Ø Factorising Cubic Functions • Use the factor theorem to find one linear factor if • is a factor Use inspection to find the quadratic factor • • • then Start with the term of the cubic Find the constant Use the term of the cubic to find the middle term of the quadratic factor Check factors using the x term of the cubic Factorise the quadratic factor if possible

The Factor Theorem Exercises Factorise the following cubics: 1. 2. 3.

The Factor Theorem

The Factor Theorem The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

The Factor Theorem SUMMARY Ø Factorising Cubic Functions • Use the factor theorem to find one linear factor if • is a factor Use inspection to find the quadratic factor • • • then Start with the term of the cubic Find the constant Use the term of the cubic to find the middle term of the quadratic factor Check factors using the x term of the cubic Factorise the quadratic factor if possible

The Factor Theorem e. g. Factorise fully Solution: We use the factor theorem to find one linear factor: Let We could try any of is not a factor is a factor So, quadratic factor The quadratic factor can be found by inspection. In this example the quadratic factor has no linear factors.