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

 Aug 5, 2023
 #1
avatar+126978 
+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

 

cool cool cool

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

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

 Aug 5, 2023
 #3
avatar+126978 
0

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

 

cool cool cool

CPhill  Aug 5, 2023
 #4
avatar+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
avatar+126978 
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

 

cool  cool cool

CPhill  Aug 5, 2023

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