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Difference between revisions of "Cubic polynomial"

A cubic polynomial is a polynomial of the form $ax^3+bx^2+cx+d=0$. A cubic polynomial is a polynomial of degree 3. A univariate cubic polynomial has the form . An equation involving a cubic polynomial is called a cubic equation. A closed-form solution known as the cubic formula exists for the solutions of an arbitrary cubic equation.

Solving a cubic

If the leading coefficient of the cubic is not 1, then divide both sides by the leading coefficient so it is 1. You get an equation of the form

$x^3 + ax^2 + bx + c = 0$

Now, we will make a change in variables to get rid of the $x^2$ term. If we do the substitution $x = y - \frac{a}{3}$, this does the trick. Our new equation is of the form

$y^3 + py = q$

We do another substitution $y = w - \frac{p}{3w}$, and our new equation is of the form

$w^3 + rw^{-3} = q$

We can now turn this into a quadratic in terms of $w^3$, solve for $w$, and then solve for $y$ and finally $x$.

Because cubic polynomials have an odd degree, their end behaviors go in opposite directions, and therefore all cubic polynomials must have at least one real root.