Factorise f(x)=3x^3+8x^2-20x-16 completely
First we need to use the factor theorem to find a factor.f(1)=-25 so x-1 is not a factorf(-1)=9 so x+1 is not a factorf(2)=0 so x-2 is a factorSo x-2 is our starting point, we need to polynomial division f(x)/x-2.(x-2)(3x^2+14x+8)=3x^3+8x^2-20x-16 by polynomial divisionSo now need to try and factorise 3x^2+14x+83 is prime so in the brakets, (3x+a)(x+b) where a and b are to be foundFactors of 8 are 42 and 81 so either (3x+8)(x+1), (3x+1)(x+8), (3x+2)(x+4) or (3x+4)(x+2) (there are also negatives)From here you can see that (3x+2)(x+4) is correct.Therefore f(x) factorised completely is (3x+2)(x+4)(x-2).
LB
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