Roots of Complex Polynomials

[BlakeJA]

Question that I came across and that has stumped me for about a week hehe.
Let $$p(z)=z^n +i z^{n-1} - 10$$

if $$\omega_j$$ are the roots for j=1,2,...,ncompute: $$\sum_{j=1}^n \omega_j}$$

and

$$\prod_{j=1}^n \omega_j}$$

 

[Muzza]

Let's consider an easier example first. Let f(x) = x^2 + 3x + 5. If f has roots a and b, then

x^2 + 3x + 5 = f(x) = (x - a)(x - b) = x^2 + x(-a - b) + ab.

Hence a + b and ab equal what? Now generalize.

 

[tacman]

The sum and product of the roots of a complex polynomial can be computed using Vieta's formulas. These formulas state that for a polynomial of degree n with roots \omega_1, \omega_2, ..., \omega_n, the sum of the roots is equal to the coefficient of the (n-1)th power term divided by the coefficient of the first power term, and the product of the roots is equal to the constant term divided by the coefficient of the first power term.

In this case, the coefficient of the (n-1)th power term is i, and the constant term is -10. Therefore, the sum of the roots is \frac{i}{1} = i, and the product of the roots is \frac{-10}{1} = -10.

So, the sum of the roots is i and the product of the roots is -10. This information can be useful in solving for the specific values of the roots, as well as in understanding the behavior of the polynomial.