How To Find The Complex Zeros Of A Polynomial Function

Multiplicity of Roots and Complex Roots

The function P(10) = (x - v)²(x + two) has iii roots-- x = v, 10 = 5, and x = - two. Since 5 is a double root, information technology is said to have multiplicity two. In general, a office with two identical roots is said to take a zero of multiplicity two. A office with iii identical roots is said to have a goose egg of multiplicity 3, and so on.

The office P(x) = x ² + 3x + 2 has two real zeros (or roots)-- ten = - 1 and x = - ii. The function P(x) = ten ² + 4 has two complex zeros (or roots)-- ten = = twoi and x = - = - twoi . The function P(x) = x ⁱⁱⁱ -11x ² + 3310 + 45 has 1 real cipher-- x = - 1--and two complex zeros-- x = 6 + 3i and x = half-dozen - 3i .

The Cohabit Zeros Theorem

The Cohabit Zeros Theorem states:

If P(ten) is a polynomial with real coefficients, and if a + bi is a zero of P , and so a - bi is a cipher of P .

Example 1: If 5 - i is a root of P(10), what is another root? Name one real cistron.

Another root is 5 + i .
A real factor is (10 - (5 - i))(x - (5 + i)) = ((ten - 5) + i)((x - 5) - i) = (x - 5)² - i ² = x ⁱⁱ -10x + 25 + 1 = x ² - 10x + 26.

Example 2: If three + iii is a root of P(x), what is another root? Proper noun i real factor.

Another root is 3 - twoi .

A real factor is (x - (3 + 2i))(x - (3 - iii)) = ((ten - 3) - twoi)((x - 3) + iii) = (x - 3)² -fouri ² = 10 ² -6x + ix + four = x ² - 6x + 13.

Example 3 If x = four - i is a zero of P(x) = x ⁱⁱⁱ -11x ² + 4110 - 51, factor P(x) completely.

By the Conjugate Zeros Theorem, we know that x = 4 + i is a zero of P(x). Thus, (10 - (4 - i))(x - (4 + i)) = ((x - 4) + i)((ten - 4) - i) = x ² - eightx + 17 is a existent cistron of P(ten). Nosotros can divide by this factor: = x - three.

Thus, P(10) = (x - 4 + i)(10 - 4 - i)(ten - three).

The Key Theorem of Algebra

The Central Theorem of Algebra states that every polynomial function of positive degree with circuitous coefficients has at to the lowest degree 1 complex zero. For example, the polynomial role P(10) = four9 ^two + threeten - 2 has at to the lowest degree ane complex aught. Using this theorem, it has been proved that:

Every polynomial part of positive degree due north has exactly n circuitous zeros (counting multiplicities).

For example, P(10) = x ⁵ + x ³ - one is a 5^th degree polynomial function, so P(x) has exactly 5 complex zeros. P(10) = 39 ² + ivx - i + 7 is a ii^nd caste polynomial function, so P(x) has exactly 2 complex zeros.