Question

The roots of x^3-8x-3=0 are given by

Answer 1

x^3−8x−3=0 

Solution

In an equation of type xn+a1xn−1+a2xn−2+.........+an, the roots of the equation are factors of an.

Here, we have the equation as x3−8x−3=0 and hence roots are factors of 3 i.e. +1 and ±3.

It is apparent that x=3 is a root as it satisfies the equation
33−8×3−3=27−24−3=0 and hence
x−3 is a factor of x3−8x−3 and dividing latter by former

x2(x−3)+3x(x−3)+1(x−3)=(x−3)(x2+3x+1)
As such we have (x−3)(x2+3x+1)=0

And as x2+3x+1 cannot be factorized as rational factors, using quadratic formula roots are
−3±√32−4×1×12=−3±√5/2

and hence three roots of x3−8x−3=0 are 3, −3+√5/2 and −3−√5/2