How do you find the number of roots for #F(x)=x^3-10x^2+27x-12# using the fundamental theorem of algebra?
The FTOA allows us to infer that
Further analysis allows us to find the
The fundamental theorem of algebra tells you that any non-constant polynomial with Complex (possibly Real) coefficients has a zero in the Complex numbers.
A simple corollary of this is that a polynomial of degree
To show the corollary, notice that if a polynomial
#f(z) = a(z-r_1)(z-r_2)...(z-r_n)#
In our example
What else can we find out about these roots of
Notice that the coefficients of
On the other hand,
Note that since
Hence we find that
By the rational root theorem, any rational roots must be expressible in the form
That means that the only possible rational roots are:
#1, 2, 3, 4, 6, 12#
(positive since we know there are no negative zeros).
Trying each of these in turn we find
#x^3-10x^2+27x-12 = (x-4)(x^2-6x+3)#
We can solve the remaining quadratic factor using the quadratic formula to find roots:
#x = (6+-sqrt(24))/2 = 3+-sqrt(6)#
So there are