Average of roots

Relevant wiki: Vieta's Formula Problem Solving - Intermediate

Let the roots of the given equation be $a_1, a_2, a_3 \cdots a_7$ . By Vieta's formula, we have the sum of roots $a_1 + a_2 + a_3 + \cdots + a_7 = -14$ . Therefore the average of the roots is $\dfrac {-14}7 = - 2$ , which is an integer. Yes .

Given any polynomial equation $x^{n} + a_{0}x^{n-1} + a_{1}x^{n-2} + \cdots + a_{n-1} = 0$ , the coefficients can be written as symmetric functions in terms of its roots. Notably, $-a_{0} = r_{1} + r_{2} + \cdots + r_{n}$ where $[r_{1},r_{2},\cdots,r_{n}]$ are the roots of the polynomial in question.

Thus, the average value of a polynomial's roots can be found by simply dividing $a_{0}$ by the number of roots. Since the fundamental theorem of algebra guarantees that an $n$ degree polynomial will always have $n$ roots, we then only need to divide $a_{0}$ by $n$ .

Thus, we have $\frac{-a_{0}}{n} = \frac{-14}{7} = -2$ , which is an integer.