Sum of the squares of the roots

Problem

For $P(x)=x^3-2x^2+4$, compute $r_1^2 + r_2^2 + r_3^2$ using the identity $p_2=\sigma_1^2-2\sigma_2$.

Solution

The polynomial is $x^3 - 2x^2 + 0\cdot x + 4$, so $a_3=1$, $a_2=-2$, $a_1=0$. The symmetric functions: $\sigma_1 = -\frac{a_2}{a_3} = 2, \qquad \sigma_2 = \frac{a_1}{a_3} = 0.$ Applying Newton’s identity $p_2=\sigma_1^2-2\sigma_2$: $r_1^2 + r_2^2 + r_3^2 = 2^2 - 2\cdot 0 = \boxed{4}.$

Links

Topics: Complex numbers
Concepts: Newton’s identities · Vieta’s relations
Skills: Using formulae
Exercise type: Evaluating expressions