__Example 1__

Show that the polynomial has as a root the number 1, with multiplicity 3.

__Solution__

If
the polynomial becomes:

So g(y), that is
f(y+1)=f(x) has as a factor the ,
but not as a power greater than y = x -1.

Thus, 1 is a root
of f(x) with multiplicity 3.

__Example 2__

Find the polynomial of 4th degree with rational coefficients, which has as roots the numbers i and .

__Solution__

Since the desired polynomial f(x) has rational
coefficients the *Theorems 5 and 6* will apply for its irrational
and complex roots.

Therefore the numbers -i, are
two other roots the polynomial.

Hence f(x) has the following form:

where a, a non zero rational number.

After doing the proper operations we get:

and one of the asked polynomials is, for
example the one we get for a=1:

.

__Example 2__

Prove that the polynomial equation , does not have integer solutions.

__Solution__

If the equations
has integer solutions, these will be divisors of
according to *Theorem 8*, hence they will be the numbers 1, -1,
5, -5.

But if ,
then:

Thus the polynomial f(x) does not have
integer solutions.