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

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

Hint: Use Rational roots theorem.

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 $12$

For the cubic equations you can use (even if is really complicated) the Cardano Formula.

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 $n>4$