Cubic equation (polynomial)

Question

A cubic polynomial with real coefficients, $a x^3 + b x^2 + c x + d$, has either three real roots, or one real root and a pair of complex conjugate ones. If the latter happens, what is the explicit formula for this real solution, and what conditions can be placed on $a,b,c$ and $d$ to guarantee that the real root is positive?

Answer 1

For explicit formulas of the roots involving the coefficients, you may wish to consider Cardano's formula which will give it for degree 3 and 4.

Answer 2

First, you need to eliminate the second degree term. For this, make $x=y+k$ with $k\in\mathbb{R}$, replace in the equation and determine $k$ in order to eliminate the second degree term.

Done that, we have an equation of the form $$ax^{3}+bx+c=0.$$ Writing $x=u+v$ and replace in the equation, you will find a quadratic equation whose roots are given in terms of $u$ and $v$ (in particular, of $x$).

Done that, it is easy to find the ultimate root.

Answer 3

Case 1

If the derivative discriminant $ b^2 - 3 a c > 0 $

Let the roots of the derivative quadratic

$$ 3 a x^2 + 2 b x + c = 0 $$

be

$$ x_1,x_2 $$

and corresponding max/ min values be

$$ y_1, y_2 $$

If these above $y$ values have the same sign ( both positive or both negative or $ y_1 y_2 >0 $) then there is a single real root and two complex conjugates.

Draw a graph of these cubic and derivative functions to ascertain why it should be so.

If $ y_1, y_2 $ are of opposite sign or $ y_1 y_2 < 0 $ then there are three real roots,and,

if one of them is zero, two real roots among them are equal.

EDIT 1:

In terms of $ (a,b,c,d ), $

(If I made no error during algebraic simplification which please check,)

we have

$ \Delta > 0 $ for two imaginary complex roots & one real root, and, $ \Delta < 0 $ for three real roots, where

$$ 27 \Delta= $$

$$ (a c^2 ((-6 + (7 - 2 a) a) b^2 + c (-3 + a^2)^2 ) - 2 c ((-6 + 4 a) b^2 + 9 c (1 + a - a^2) ) d + 27 d^2). $$

Case 2

If the derivative discriminant $ b^2 - 3 a c < 0 $ then we have an inflection point on the x-axis.