Find the solutions to $z^3 = -8 - 7z.$ Enter the solutions, separated by commas.

maximum Aug 5, 2023

#1**+1 **

Rearrange as

z^3 + 7z + 8 = 0

By inspection , I can see that z = -1 is one root

Using synthetic division to find the remaining polynomial we have

-1 [ 1 0 7 8 ]

-1 1 -8

_______________

1 -1 8 0

The remaining polynomial is z^2 - z + 8

The other two (non-real) roots are

[ 1 + sqrt ( 1 - 4(1) (8) ] / 2 = [ 1 - i sqrt ( 31)] / 2

And [ 1 + i sqrt (31) ] / 2

CPhill Aug 5, 2023

#2

#4**+1 **

I found this page on it: https://brilliant.org/wiki/cardano-method/. If your intrested you can go check it out. This offers a solution to the full cubic by first depressing it and then solving that.

jonathanldong
Aug 5, 2023