Advanced algebra... sum of five roots?

Question

What is the sum of the five roots of the equation

x^5 – 4x^3 5x – 11 = 0

i need more than just the answer.... how do i find it? something about

-b/a but that didn't work......

Answer 1

Given the fifth degree equation

ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0

or,  dividing by a:

x^5 + (b / a)x^4 + (c / a)x^3 + (d / a)x^2 + (e / a)x + (f / a) ...(1)

the sum of the roots is - b / a.

Let the roots of the equation be p, q, r, s and t.

Then the equation can be factorised (even if you can't find the factors) as:

(x - p)(x - q)(x - r)(x - s)(x - t) = 0

When you multiply these brackets out, the coefficient of x is:

- (p + q + r + s + t).

Comparing this coefficient with the one in (1):

p + q + r + s + t = - b / a.

For the equation given, as there is no x^4 term, the sum of the roots is:

- b / a

= 0.