# How do you find the number of roots for #F(x)=x^3-10x^2+27x-12# using the fundamental theorem of algebra?

##### 1 Answer

The FTOA allows us to infer that

Further analysis allows us to find the

#### Explanation:

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)#

where

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