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:

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:

For example, P(x) = x⁵ + x³ - 1 is a 5^th degree polynomial function, so P(x) has exactly 5 complex zeros. P(x) = 3ix² + 4x - i + 7 is a 2^nd degree polynomial function, so P(x) has exactly 2 complex zeros.

The second part of the Fundamental Theorem of Algebra is more useful in Algebra 1; in fact, the first part will rarely be used in problems.