Sum and product of the roots

Problem

Compute the sum and product of the four roots of $x^4 - x^3 + 2x - 5 = 0$ in $\mathbb{C}$.

Solution

The polynomial is $x^4 - x^3 + 0\cdot x^2 + 2x - 5$, so $a_4=1$, $a_3=-1$, $a_0=-5$. Using the generalised Vieta’s relations ($n=4$): $\sigma_1 = r_1+r_2+r_3+r_4 = -\frac{a_3}{a_4} = -\frac{-1}{1} = \boxed{1},$ $\sigma_4 = r_1 r_2 r_3 r_4 = (-1)^4\frac{a_0}{a_4} = \frac{-5}{1} = \boxed{-5}.$

Links

Topics: Complex numbers
Concepts: Vieta’s relations · Fundamental theorem of algebra
Methods: Generalised Vieta
Skills: Using formulae
Exercise type: Evaluating expressions