Complex Coefficients and Zeros
Most of the polynomials
we deal with have real coefficients.
However, there are many more types of polynomials out there in the world!
What happens if some or all of the coefficients are complex
numbers? Then we have the following theorem:
Theorem If a polynomial has degree
greater than zero and complex coefficients, then it must have at least
one complex root.
Note that in this context, "complex" includes the case of real numbers. Thus,
a polynomial with real coefficients must have at least one complex root
a + bi - but
b could be zero, in which case the
root is actually real. In general, all combinations are possible:
Examples
-
f(x) = x2 + 1
has no real roots, but it does have the complex roots
i and
-i.
So here we have real coefficients and complex, non-real, roots.
-
f(x) = x2 - 1
has only real roots:
1 and
-1.
So here we have real coefficients and real roots.
-
f(x) = ix2 + 1
has 2 complex roots:
±(1/sqrt(2))(1+i).
-
f(x)
= x2 + (-1 + i)x - i
has complex coefficients and both a real root and a complex, non-real root:
1 and
-i.