+0

# Polynomial roots

0
36
5
+817

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

Aug 5, 2023

#1
+129690
+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

Aug 5, 2023
edited by CPhill  Aug 5, 2023
#2
+32
+1

I think this is a type of equation called a depressed cubic.

Aug 5, 2023
#3
+129690
0

You're correct.....it even LOOKS depressing  (LOL!!!)

CPhill  Aug 5, 2023
#4
+32
+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
edited by jonathanldong  Aug 5, 2023
#5
+129690
0

Thanks for the reference , jonathan....I knew there was a method to solve such a thing, but as I remember, it was a little involved

CPhill  Aug 5, 2023