solve for x......x^4-8x^3-375=0

Question

I am looking for the roots of

   f(x) = x^4 - 8x^3 - 375

Answer 1

Use Newton's Method:

f'(x)=4x³-24x²;

x_n+1=x_n-f(x_n)/f'(x_n); start with x₀=1, then:

x₁=x₀-f(x₀)/f'(x₀)=1-(1-8-375)/(4-24)=-181/10=-18.1;

x₂=-13.211...; x₃=-9.592...; x₄=-6.952...; x₅=-5.095...; x₆=-3.917...;

x₇=-3.356...; x₈=-3.227...; x₉=-3.221...; ...x_n=-3.221062119 approx is one solution.

A rough graph of f(x) shows another solution near x=9, so let x₀=9:

after several iterations x_n=8.591352743 approx is another solution.

There are no other real zeroes for x; the complex zeroes can be found by dividing by the discovered factors x+3.221062119 and x-8.591352743 to obtain the quadratic quotient, which can then be solved using the quadratic formula.