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**“Teach A Level Maths” Vol. 1: AS Core Modules**

21:The Factor Theorem © Christine Crisp

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**A simple quadratic function can be factorised by inspection e. g**

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

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e.g. If is a factor This time there is no remainder For a polynomial function , the factor theorem says that: if then is a factor

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**Factorising a Cubic Function**

e.g.1 Find a linear factor of Solution: Let Try x = 1: is a factor Once we have found one factor of a cubic, the other factor, which is quadratic, can be found by long division. Sometimes this quadratic factor will also factorise

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e.g.2 Factorise fully Solution: We use the factor theorem to find one linear factor: Let is not a factor is a factor So, quadratic factor The quadratic factor can be found by long division.

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**The quadratic factor has no real roots as b2–4ac<0**

x3+ x2 -3x2+ 5x -3x2– 3x 8x+8 8x+8 We have The quadratic factor has no real roots as b2–4ac<0

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1 real root at x = –1

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**SUMMARY Factorising Cubic Functions**

Use the factor theorem to find one linear factor if then is a factor Use long division to find the quadratic factor Factorise the quadratic factor if possible

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Exercises Factorise the following cubics: 1. 2. 3.

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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.

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**SUMMARY Factorising Cubic Functions**

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

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**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.

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