SOLUTION: I have to find all roots of X^6 + 9X^3 + 8 = 0. I first let Y=X^3, then my equation became Y^2 + 9Y + 8 = 0. Solving for Y, I have Y = -8 OR -1. So I have two possible roots as X

Question 231979: I have to find all roots of X^6 + 9X^3 + 8 = 0.
I first let Y=X^3, then my equation became Y^2 + 9Y + 8 = 0. Solving for Y, I have Y = -8 OR -1.
So I have two possible roots as X = -2 and X = -1.
Since the highest power in the equation is 6, there are four other roots.
How do you find the other roots?
Thanks,
Alan
Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!

I took a direct approach and factored it into

is a solution

I need the other 2 roots now. I proceed to cheat and
apply what I know about complex roots, namely that
they are equally spaced aound the complex plane. That means
the 3 roots are 120 degrees apart. One complex root is
30 degrees from the vertical in 1st quadrant, the other is
30 degrees from vertical in 4th quadrant. The components of
the complex roots are
(1st quadrant) and
(4th quadrant).
I do the same for

The complex roots are
(1st quadrant) and
(4th quadrant
I don't know an easy way to prove that's right, but
I'm pretty sure it is

RELATED QUESTIONS
I'm suppose to writ these equation in standard form using only integers. so I have... (answered by Alan3354)

I know how to graph and substitute but I just can't figure out this "if" statements, I... (answered by josgarithmetic,ikleyn)

For the equation 9 x ^3 - 9 y ^2 = -6 answer the following questions. Is the equation... (answered by solver91311)

Write the equation of the line passing through (3,-2), but perpendicular to y=3. I... (answered by Alan3354,solver91311)

Use the substitution y=3^x to find the values of x which satisfy the equation... (answered by Alan3354)

I have a substitution problem that I would like help with, please (I have a test... (answered by Alan3354)

How would you find the inverse of one to one function f(x)=9x/x-8? I have worked it... (answered by Alan3354)

I have tried to figure out how to properly graph this problem however, when I looked at... (answered by Fombitz,algebrapro18)

how do I solve this equation? I have to find the real roots for {{{y^(-2)-2/y... (answered by vleith)