+817 Find the solutions to $z^3 = -8 - 7z.$ Enter the solutions, separated by commas.
+129771 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
+32 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.
