# How do you find the number of roots for #f(x) = x^3 – 75x + 250# using the fundamental theorem of algebra?

##### 1 Answer

The FTOA tells us that there are

Further investigation shows us that they are

#### Explanation:

The fundamental theorem of algebra tells you that any non-constant polynomial in one variable with Complex (possibly Real) coefficients has a zero which is a Complex (possibly Real) number.

A corollary of this, often stated as part of the FTOA is that a polynomial in one variable of degree

So in our example:

#f(x) = x^3-75x+250#

is of degree

**Bonus**

What else can we find out about these zeros?

By the rational root theorem, any rational zeros of

That means that the only possible rational zeros are positive and negative factors of

#+-1# ,#+-2# ,#+-5# ,#+-10# ,#+-25# ,#+-50# ,#+-125# ,#+-250#

We find:

#f(-10) = -1000+750+250 = 0#

So

#x^3-75x+250#

#=(x+10)(x^2-10x+25)#

#=(x+10)(x-5)(x-5)#

So the zeros are