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.