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a + bi
is a zero, then a  bi
is also a zero. Understanding complex zeros is crucial for fully solving polynomial equations and for understanding the nature of polynomials. Polynomial functions can have complex zeros when their solutions involve imaginary numbers. Complex zeros always occur in conjugate pairs if the polynomial has real coefficients. This means if a + bi
is a zero, then a  bi
is also a zero. Understanding complex zeros is crucial for fully solving polynomial equations and for understanding the nature of polynomials.
Here are ten examples of polynomial functions and their complex zeros:
f(x) = x^2 + 1
. Zeros: i, i
f(x) = x^2 + 4
. Zeros: 2i, 2i
f(x) = x^2 + 9
. Zeros: 3i, 3i
f(x) = x^2  2x + 5
. Zeros: 1 + 2i, 1  2i
f(x) = x^2 + 6x + 10
. Zeros: 3 + i, 3  i
f(x) = x^2  4x + 13
. Zeros: 2 + 3i, 2  3i
f(x) = x^2 + x + 1
. Zeros: (1/2) + (sqrt(3)/2)i, (1/2)  (sqrt(3)/2)i
f(x) = x^4 + 4
. Zeros: sqrt(2)/2 + (sqrt(2)/2)i, sqrt(2)/2 + (sqrt(2)/2)i, sqrt(2)/2  (sqrt(2)/2)i, sqrt(2)/2  (sqrt(2)/2)i
f(x) = x^4  10x^2 + 9
. Zeros: sqrt(3), sqrt(3), 3i, 3i
f(x) = x^3  3x^2 + 4
. Zeros: 2, 1 + i, 1  i
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