How do you find the roots for #f(x) = 17x^15 + 41x^12 + 13x^3 - 10# using the fundamental theorem of algebra?
1 Answer
The FTOA does not help you find the zeros - it only tells you that this polynomial of degree
Explanation:
By roots, I will assume you mean zeros, i.e. values of
The so called fundamental theorem of algebra (FTOA) is neither fundamental nor a theorem of algebra, but what it does tell you is that any non-zero polynomial in one variable with Complex coefficients has a zero in
A simple corollary of this - often stated as part of the FTOA - is that a polynomial in one variable of degree
In our example,
The FTOA does not help you actually find the zeros.
Bonus
What else can we find out about the zeros of this
Note that the coefficients of
Note
So the positive Real zero is in
#f(-x) = -17x^5+41x^12-13x^3-10#
has two changes of sign, so
We find:
#f(-1) = -17+41-13-10 = 1 > 0#
So there is a Real zero in
Any other zeros will occur in Complex conjugate pairs, since all of the coefficients of
Note that all of the degrees are multiples of
#g(t) = 17t^5+41t^4+13t-10#