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.
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.