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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.


Note: The above method will work for any cubic, but the cubic polynomials in these problems have special tricks to solve them.

2013 AIME I Problems/Problem 5 The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

2014 AIME I Problems/Problem 9 Let $x_1<x_2<x_3$ be the three real roots of the equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.

2010 AIME II Problems/Problem 7 Let $P(z)=z^3+az^2+bz+c$, where a, b, and c are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $|a+b+c|$.

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