What polynomial function has zeros 1,-4 and -3i?

Question

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Answer 1

Assuming the polynomial has real coefficients, the complex conjugate of a complex zero must also be a zero.  So far the zeros are

 

1

-4

-3i

3i

 

If the polynomial is of degree 4,

 

P(x) = A(x - 1)(x + 4)(x + 3i)(x - 3i)

 

A ≠ 0

 

Note that (x + 3i)(x  - 3i) = x² + 9

 

P(x) = A(x - 1)(x + 4)(x² + 9)

 

You can expand this into standard form if you wish.  Note that all of the coefficients are real, even though there are two complex zeros.