What is the simplest polynomial with real coefficients and zeros #7#, #-11# and #2+8i# ?

1 Answer
George C.
Dec 15, 2017

#f(x) = x^4 - 25x^2 + 580x - 5236#

Explanation:

Given that the polynomial has zeros #7#, #-11# and #2+8i#, it has factors:

#(x-7)#, #(x+11)# and #(x-2-8i)#

In order to have real coefficients, it must also have #2-8i# as a zero and #(x-2+8i)# as a factor.

So the simplest polynomial would be:

#f(x) = (x-7)(x+11)(x-2-8i)(x-2+8i)#

#color(white)(f(x)) = (x^2+4x-77)(x+11)((x-2)^2-(8i)^2)#

#color(white)(f(x)) = (x^2+4x-77)(x^2-4x+4+64)#

#color(white)(f(x)) = (x^2+4x-77)(x^2-4x+68)#

#color(white)(f(x)) = x^4 - 25x^2 + 580x - 5236#

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