Question

Factorise: x^3-5x^2+2x+8

Answer 1

Explanation:

As the coefficient of x^3 in x^3−5x^2+2x+8=0 is 1,

One of the roots must be factor of 8/1=8 i.e. it can be ±1.±2,±4or±8.

As sum of coefficients (1−5+2+8=6≠0) is not zero, it is evident that 1 is not the root of x^3−5x^2+2x+8=0.

But −1 is a root as −1−5−2+8=0 and hence as per factor theorem (x+1)is a factor of x^3−5x^2+2x+8=0

Dividing x^3−5x^2+2x+8 by (x+1) we get

x^3−5x^2+2x+8 = x^2.(x+1)−6x(x+1)+8(x+1) = (x+1)(x2−6x+8)

now we can further factorize x^2−6x+8 by splitting middle term

x^2−6x+8=x^2−4x−2x+8=x(x−4)−2(x−4)=(x−2)(x−4)

Hence, roots of x3−5x2+2x+8=0 are {−1,2,4}.

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