21:The Factor Theorem © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

The Factor Theorem Module C1 AQA Edexcel OCRMEI/OCR Module C2 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

The Factor Theorem A simple quadratic function can be factorised by inspection e.g. Consider The factors are and From the factors, we can see that the zeros of the functions are 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, the factor theorem says that: if then is a factor

The Factor Theorem Factorising a Cubic Function Solution: Let e.g.1 Find a linear factor of 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 Try a = 1:

The Factor Theorem These numbers suggest that could be a factor Solution: We use the factor theorem to find one linear factor: e.g.2 Factorise fully The linear factor can only contain numbers that are factors of 8. Let We could try any of is not a factor is a factor So, 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 haveSo, 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 Use the factor theorem to find one linear factor Use inspection to find the quadratic factor 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 Factorise the quadratic factor if possible Check factors using the x term of the cubic  Factorising Cubic Functions if then is a factor

The Factor Theorem Exercises Factorise the following cubics:

The Factor Theorem

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The Factor Theorem SUMMARY Use the factor theorem to find one linear factor Use inspection to find the quadratic factor 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 Factorise the quadratic factor if possible Check factors using the x term of the cubic  Factorising Cubic Functions if then is a factor

The Factor Theorem Solution: We use the factor theorem to find one linear factor: e.g. Factorise fully 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.