2A simple quadratic function can be factorised by inspection e. g A simple quadratic function can be factorised by inspection e.g. ConsiderThe factors areFrom the factors, we can see that the roots of the function areandSo,Reversing this process enables us to factorise cubics ( and polynomials of higher degrees )
3e.g. Ifis a factorThis time there is no remainderFor a polynomial function , the factor theorem says that:if then is a factor
4Factorising a Cubic Function e.g.1 Find a linear factor ofSolution: LetTry x = 1:is a factorOnce 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
5e.g.2 Factorise fullySolution: We use the factor theorem to find one linear factor:Letis not a factoris a factorSo,quadratic factorThe quadratic factor can be found by long division.
6The quadratic factor has no real roots as b2–4ac<0 x3+ x2-3x2+ 5x-3x2– 3x8x+88x+8We haveThe quadratic factor has no real roots as b2–4ac<0
11The 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.
12SUMMARY Factorising Cubic Functions Use the factor theorem to find one linear factorUse inspection to find the quadratic factorStart with the term of the cubicFind the constantUse the term of the cubic to find the middle term of the quadratic factorFactorise the quadratic factor if possibleCheck factors using the x term of the cubicFactorising Cubic Functionsif then is a factor
13Solution: We use the factor theorem to find one linear factor: e.g. Factorise fullyLetWe could try any ofis not a factoris a factorSo,quadratic factorThe quadratic factor can be found by inspection.In this example the quadratic factor has no linear factors.