# How do you determine the number of complex roots of a polynomial of 3 degrees?

Oct 30, 2015

See explanation...

#### Explanation:

Suppose we are given $f \left(x\right) = a {x}^{3} + b {x}^{2} + c x + d$ with $a > 0$

The slope of the tangent of the graph of $f \left(x\right)$ at any $x$ is given by the derivative:

$f ' \left(x\right) = 3 a {x}^{2} + 2 b x + c$

Using the quadratic formula, the cubic has turning points where $f ' \left(x\right) = 0$, which is when

$x = \frac{- 2 b \pm \sqrt{4 {b}^{2} - 12 a c}}{6 a} = \frac{- b \pm \sqrt{{b}^{2} - 3 a c}}{3 a}$

If ${b}^{2} - 3 a c < 0$ then $f ' \left(x\right) = 0$ only has Complex roots, so $f \left(x\right)$ has no Real turning points and therefore $f \left(x\right) = 0$ has exactly one Real root and two non-Real Complex roots.

If ${b}^{2} - 3 a c = 0$ then $f ' \left(x\right) = 0$ has one repeated Real root, so $f \left(x\right)$ has an inflection point, but is strictly monotonically increasing. So again $f \left(x\right) = 0$ has exactly one Real root and two non-Real Complex roots, unless the inflection point is itself a root. If $f \left(- \frac{b}{3 a}\right) = 0$ then this is a threefold Real root and there are no non-Real Complex roots.

If ${b}^{2} - 3 a c > 0$ then $f ' \left(x\right) = 0$ has two distinct Real roots:

${x}_{1} = \frac{- b - \sqrt{{b}^{2} - 3 a c}}{3 a}$

${x}_{2} = \frac{- b + \sqrt{{b}^{2} - 3 a c}}{3 a}$

We know that $f \left({x}_{2}\right) < f \left({x}_{1}\right)$, but there are still several possibilities:

(1) $f \left({x}_{1}\right) < 0$, $f \left({x}_{2}\right) < 0$

$f \left(x\right) = 0$ has one Real root in $\left({x}_{2} , \infty\right)$ and two non-Real Complex roots.

(2) $f \left({x}_{1}\right) = 0$, $f \left({x}_{2}\right) < 0$

$f \left(x\right) = 0$ has a repeated Real root at $x = {x}_{1}$, and another Real root in $\left({x}_{2} , \infty\right)$.

(3) $f \left({x}_{1}\right) > 0$, $f \left({x}_{2}\right) < 0$

$f \left(x\right) = 0$ has three distinct Real roots in $\left(- \infty , {x}_{1}\right)$, $\left({x}_{1} , {x}_{2}\right)$ and $\left({x}_{2} , \infty\right)$.

(4) $f \left({x}_{1}\right) > 0$, $f \left({x}_{2}\right) = 0$

$f \left(x\right) = 0$ has one Real root in $\left(- \infty , {x}_{1}\right)$ and a repeated Real root at $x = {x}_{2}$.

(5) $f \left({x}_{1}\right) > 0$, $f \left({x}_{2}\right) > 0$

$f \left(x\right) = 0$ has one Real root in $\left(- \infty , {x}_{1}\right)$ and two non-Real Complex roots.