The function
*P*(*x*) = (*x* - 5)^{2}(*x* + 2)
has 3 roots--
*x* = 5
,
*x* = 5
, and
*x* = - 2
. Since 5 is a double root, it is said to have multiplicity two. In
general, a function with two identical roots is said to have a zero of
multiplicity two. A function with three identical roots is said to have a
zero of multiplicity three, and so on.

The function
*P*(*x*)
=
*x*
^{2} + 3*x* + 2
has two
real zeros (or roots)--
*x* = - 1
and
*x* = - 2
. The function
*P*(*x*) = *x*
^{2} + 4
has two
complex zeros (or roots)--
*x* = = 2*i*
and
*x* = - = - 2*i*
. The function
*P*(*x*) = *x*
^{3} -11*x*
^{2} + 33*x* + 45
has one real zero--
*x* = - 1
--and two complex zeros--
*x* = 6 + 3*i*
and
*x* = 6 - 3*i*
.

The Conjugate Zeros Theorem states:

IfP(x) is a polynomial with real coefficients, and ifa+biis a zero ofP, thena-biis a zero ofP.

*Example 1*: If
5 - *i*
is a root of
*P*(*x*)
, what is another root? Name
one real factor.

Another root is
5 + *i*
.

A real factor is
(*x* - (5 - *i*))(*x* - (5 + *i*)) = ((*x* - 5) + *i*)((*x* - 5) - *i*) = (*x* - 5)^{2} - *i*
^{2} = *x*
^{2} -10*x* + 25 + 1 = *x*
^{2} - 10*x* + 26
.

*Example 2*: If
3 + 2*i*
is a root of
*P*(*x*)
, what is another root? Name
one real factor.

Another root is
3 - 2*i*
.

A real factor is
(*x* - (3 + 2*i*))(*x* - (3 - 2*i*)) = ((*x* - 3) - 2*i*)((*x* - 3) + 2*i*) = (*x* - 3)^{2} -4*i*
^{2} = *x*
^{2} -6*x* + 9 + 4 = *x*
^{2} - 6*x* + 13
.

*Example 3* If
*x* = 4 - *i*
is a zero of
*P*(*x*) = *x*
^{3} -11*x*
^{2} + 41*x* - 51
,
factor
*P*(*x*)
completely.

By the Conjugate Zeros Theorem, we know that
*x* = 4 + *i*
is a zero of
*P*(*x*)
.
Thus,
(*x* - (4 - *i*))(*x* - (4 + *i*)) = ((*x* - 4) + *i*)((*x* - 4) - *i*) = *x*
^{2} - 8*x* + 17
is a real factor of
*P*(*x*)
. We can divide by this factor:
= *x* - 3
.

Thus,
*P*(*x*) = (*x* - 4 + *i*)(*x* - 4 - *i*)(*x* - 3)
.

The Fundamental Theorem of Algebra states that every polynomial function of
positive degree with complex coefficients has at least one complex zero. For
example, the polynomial function
*P*(*x*) = 4*ix*
^{2} + 3*x* - 2
has at least one
complex zero. Using this theorem, it has been proved that:

Every polynomial function of positive degreeFor example,nhas exactlyncomplex zeros (counting multiplicities).