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Is there any condition based on the coefficients of terms that guarantees all real solutions to a general cubic polynomial? e.g. $$ax^3+bx^2+cx+d=0\, ?$$

If not, are there methods rather than explicit formula to determine it?

Thank You.

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how you mean guarantee? what conditions you need that all solutionas are real ? –  Dominic Michaelis Apr 4 '13 at 8:52
    
I think this is quite directly related to the calculations of Galois groups of polynomials, at least in the special case of a third degree polynomial. So per chance this could be tagged galois theory as well? Thanks. –  awllower Apr 4 '13 at 8:58

2 Answers 2

up vote 7 down vote accepted

Wikipedia says it's the following expression: $$ 18abcd-4b^3d + b^2c^2 -4ac^3 -27a^2d^2 $$ If it's positive there are three real roots. If it's negative there's one real and two (conjugate) imaginary roots. If it's equal to zero, there are fewer than three distinct roots, but they are all real.

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+1 Dang, beat me to it ;-) –  Fixed Point Apr 4 '13 at 8:56
    
'Three non-distinct' implies that they are same always, but that is only true if the polynomial is $x^3$. In other cases only $2$ roots are equal. –  lsp Apr 4 '13 at 9:02
    
@lsp As I see it, "three non-distinct roots" means "three roots of which it is not true that they are distinct". In my opinion that covers both the case one double root plus one single root and the case one triple root. –  Arthur Apr 4 '13 at 9:06
    
@Arthur If that was what you meant, then it is correct. :) –  lsp Apr 4 '13 at 9:07
    
But this usage as it stands now is a little confusing; hence, if this is not too troublesome, might you consider explain more in the answer? Thanks in advance. –  awllower Apr 4 '13 at 10:32

Discriminant, $D = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2$

If $D < 0$, then the polynomial has two complex roots,

If $D = 0$, then there are three real roots, and two of them are definitely equal,

If $D > 0$, then there are three distinct real roots.

Refer http://en.wikipedia.org/wiki/Casus_irreducibilis for more information.

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Note: if $D = 0$ all three of the roots might be equal too, as in the polynomial $x^3$ –  Arthur Apr 4 '13 at 8:58
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@Arthur Your comment contradicts your own answer... Consider a little modification? –  awllower Apr 4 '13 at 8:59
    
@awllower Nope, I say three non-distinct roots. –  Arthur Apr 4 '13 at 9:00
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$3$ non-distinct roots does not include the case of $2$ roots being equal. –  lsp Apr 4 '13 at 9:03

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