Prove general solution for y^3+py+q=0 given a delta

Palensya

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I've been given the following problem and I can't seem to figure out the solution, would appreciate any help and direction to solution !

Let's look at the equation : y3 + py + q = 0 (*)
We shall define Delta (or d in short) d = 4p3 + 27q2
Prove that :
a) If d > 0 , then (*) has a single solution .

b) If d = 0 , and at least one of the coefficients (p , q) =/= 0, then (*) has 2 solutions .

c) If d < 0 , then (*) has 3 solutions .


Thank you very much in advance !


 
I've been given the following problem and I can't seem to figure out the solution, would appreciate any help and direction to solution !

Let's look at the equation : y3 + py + q = 0 (*)
We shall define Delta (or d in short) d = 4p3 + 27q2
Prove that :
a) If d > 0 , then (*) has a single solution .

b) If d = 0 , and at least one of the coefficients (p , q) =/= 0, then (*) has 2 solutions .

c) If d < 0 , then (*) has 3 solutions .
Thank you very much in advance !

Have you looked into Cardano's solution? For a quick reference go to:

https://en.wikipedia.org/wiki/Cubic_function
 
I've been given the following problem and I can't seem to figure out the solution, would appreciate any help and direction to solution !

Let's look at the equation : y3 + py + q = 0 (*)
We shall define Delta (or d in short) d = 4p3 + 27q2
Prove that :
a) If d > 0 , then (*) has a single solution .

b) If d = 0 , and at least one of the coefficients (p , q) =/= 0, then (*) has 2 solutions .

c) If d < 0 , then (*) has 3 solutions .

If you want help in proving this, we'll need to know your context. What facts have you learned, that they will expect you to be able to use in your proof?

Also, since you have apparently tried something, please show us your attempt, so we can see how far you are from something useful.

If you haven't done so already, please read this.
 
Have you looked into Cardano's solution? For a quick reference go to:

https://en.wikipedia.org/wiki/Cubic_function

Looked in this reference, a lot of interesting information, but like Dr.Peterson above me said it didn't match the context and facts I've learned and need to prove with !

If you want help in proving this, we'll need to know your context. What facts have you learned, that they will expect you to be able to use in your proof?

Also, since you have apparently tried something, please show us your attempt, so we can see how far you are from something useful.

If you haven't done so already, please read this.

It's supposed to use Rolle's theorem, Lagrange's theorem for derivatives and Cauchy's theorem for derivatives .

I'm not sure if I am capable of translating my attempts to English (since I study in a different language) but I can try to explain where I got stuck .


For a) I showed that for d > 0 follows p < 0 . Which, by the way, should actually give 3 solutions if i'm correct, and there might be a mistake in the question or in my understanding ?

It got me really confused .

For sake of saving time I assumed that for d > 0 (=> p < 0 ) we get 3 solutions, I proved using Weierstrass's intermediate value theorem to prove there exists a solution for f(y)=0 , but I can't seem to disprove that the number of solutions that satisfy this equation is > 3 and < 3 .
 
There is a rather cumbersome proof based on the fundamental theorem of algebra that involves no calculus at all. Would that be pertinent?
 
There is a rather cumbersome proof based on the fundamental theorem of algebra that involves no calculus at all. Would that be pertinent?

I'd love to look at it , It might give me some direction :D
 
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