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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) = x2 + 3x + 2 has two
real zeros (or roots)--x = - 1 and x = - 2. The function P(x) = x2 + 4 has two
complex zeros (or roots)--x = 
= 2i and x = - = - 2i. The function P(x) = x3 -11x2 + 33x + 45 has one real zero--x = - 1--and two complex zeros--x = 6 + 3i
and x = 6 - 3i.
The Conjugate Zeros Theorem states:
If P(x) is a polynomial with real coefficients, and if a + bi is a zero of P, then a - bi is a zero of P.
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 - i2 = x2 -10x + 25 + 1 = x2 - 10x + 26.
Example 2: If 3 + 2i is a root of P(x), what is another root? Name
one real factor.
Another root is 3 - 2i.
A real factor is (x - (3 + 2i))(x - (3 - 2i)) = ((x - 3) - 2i)((x - 3) + 2i) = (x - 3)2 -4i2 = x2 -6x + 9 + 4 = x2 - 6x + 13.
Example 3 If x = 4 - i is a zero of P(x) = x3 -11x2 + 41x - 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) = x2 - 8x + 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) = 4ix2 + 3x - 2 has at least one
complex zero. Using this theorem, it has been proved that:
Every polynomial function of positive degree n has exactly n complex zeros (counting multiplicities).For example, P(x) = x5 + x3 - 1 is a 5th degree polynomial function, so P(x) has exactly 5 complex zeros. P(x) = 3ix2 + 4x - i + 7 is a 2nd degree polynomial function, so P(x) has exactly 2 complex zeros.
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