Algebra II: Polynomials: Complex Zeros and the Fundamental Theorem of Algebra

Multiplicity of Roots and Complex Roots

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

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

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 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.

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