What is a fourth degree polynomial function with real coefficients that has 2, -2 and -3i as zeros?

1 Answer
Feb 6, 2016

Answer:

#f(x)=x^4+5x^2-36#

Explanation:

If #f(x)# has zeroes at #2 and -2#
it will have #(x-2)(x+2)# as factors.

If #f(x)# has a zero at #-3i#
then #(x+3i)# will be a factor
and we will need to use a fourth factor to "clear" the imaginary component from the coefficients.
(Remember we were told the polynomial was of degree 4 and has no imaginary components).

The most obvious fourth factor would be the complex conjugate of #(x+3i)#, namely #(x-3i)#
and since #(x+3i)(x-3i) = x^2+9#

#f(x)=(x-2)(x+2)(x^2+9)#
which can be expanded to:
#f(x)=x^4+5x^2-36#