# Is there a general method for solving a cubic polynomial? [duplicate]

This question already has an answer here:

I'm doing a course in linear algebra at the moment, and whenever I need to find the eigenvalues of a 3x3 matrix, I'm faced with the issue that I don't know a general method for solving a cubic equation, or I don't remember enough from high school algebra to be able to solve any given cubic equation.

For instance, if I have the characteristic equation

$$-\lambda^3+8\lambda^2-19\lambda+12=0$$

then I can't factor it easily because 19 is a prime number, but a computer was able to somehow find it's solutions for me.

Is there a general, algorithmic method for solving any given cubic equation without factorization? If not, how would you solve an equation like the above?

And is there a similar kind of general method for solving quartic or higher degree polynomials?

## marked as duplicate by Ross Millikan algebra-precalculus StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 4 '14 at 14:40

• for higher polynomials of degree more than $4$ answer is NO... – user87543 Oct 4 '14 at 14:36
• – Paracosmiste Oct 4 '14 at 14:36
• If you search the site for Cardano or cubic this is answered many times. – Ross Millikan Oct 4 '14 at 14:41
• 19 being prime is not important. You should worry more about the constant term. For example, the quadratic equation $X^2-pX+7(p-7)$ has solutions $7, p-7$, for any $p$, prime or not. – Slade Oct 4 '14 at 15:05
• But $X^2 - aX + p$ usually doesn't have solutions, by the rational root theorem. – Slade Oct 4 '14 at 15:06

The R.R. Theorem says the rational roots of $-\lambda^3 + 8\lambda^2 -19\lambda + 12$ have to be of the form $$\pm \frac{a}{1}$$ where $a$ is a factor of $12$
There are results in Agebra that show there are no such closed formulas for higher degree equations. This depends on the non-solvability property of permutations groups $S_n$ for $n>4$