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

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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 roots of the function are So, Reversing this process enables us to factorise cubics ( and polynomials of higher degrees )

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

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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 long division. Sometimes this quadratic factor will also factorise Try x = 1:

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The Factor Theorem Solution: We use the factor theorem to find one linear factor: e.g.2 Factorise fully 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 Factor Theorem We have x 3 + x 2 -3x 2 + 5x -3x 2 – 3x 8x+8 The quadratic factor has no real roots as b 2 –4ac<0

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The Factor Theorem 1 real root at x = –1

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The Factor Theorem SUMMARY Use the factor theorem to find one linear factor Use long division to find the quadratic factor Factorise the quadratic factor if possible Factorising Cubic Functions if then is a factor

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

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The Factor Theorem

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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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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

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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.

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