Question

Use the Fundamental Theorem of Algebra to explain how many roots your expression can have. How many real roots and how many complex roots are possible? (x+3)^10

Answer 1

We have the relation:

(x + 3)^10

As the degree of the polynomial is 10, we know that we can have 10 roots.

The theorem says that this has at least one complex root.

As you know, this is actually trivial, because for example, 4 can be a complex number with an imaginary part equal to zero.

In this case, the polynomial is only zero if we have x = 3.

Then we have 10 roots that are equal to x = 3.

and those 10 roots can be real, or we can have 10 roots equal to 3 + 0*i, that is a complex number.