If a polynomial function with rational coefficients has the zeros -1+i, sqrt5, what are the additional zeros?

1 Answer
Shell
Sep 18, 2016

The additional zeros are #(-1-i)# and #-sqrt5#.

Explanation:

Complex zeros always occur in pairs in the form of complex conjugates #a+-bi#.

The complex conjugate of #(-1+i)# is #(-1-i)#.

Similarly, zeros containing square roots must also come in pairs. If the polynomial has a zero of #sqrt5#, it must also have a zero of #-sqrt5#.

Think about the quadratic formula #x=frac{-b+-sqrt(b^2-4ac)}{21}#.

If the discriminant #b^2-4ac# is negative, there will be two complex zeros because of the #+-# signs before the square root. These are complex conjugates.

If the discriminant is positive but not a perfect square, there will also be two zeros of the form #+-sqrt#

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