How do you find the number of roots for #f(x) = 4x^5 – 2x – 15# using the fundamental theorem of algebra?
1 Answer
Explanation:
The Fundamental Theorem of Algebra tells us that any non-constant polynomial equation with Complex (possibly Real) coefficients has at least one Complex (possibly Real) zero.
A simple corollary of this is that a polynomial of degree
To show the corollary: Suppose
So in the case of
Bonus
What else can we find out about these zeros?
First note that the sequence of coefficients of
In fact, we can see that
If we reverse the signs of the coefficients of the terms of odd degree, then all are negative. So
Since the coefficients of
So we must have two pairs of Complex conjugate zeros and one positive Real zero.