What is the polynomial function #f# of least degree that has rational coefficients, a leading coefficient of 2, and the zeros #0, 3+4i#?

1 Answer
Mar 10, 2018

#f(x) = 2x^3-12x^2+50x#

Explanation:

If a polynomial has rational coefficients, then any radical or complex zeros will occur in conjugate pairs.

So if #3+4i# is a zero then so is #3-4i#

A polynomial has a zero #x=a# if and only if it has a factor #(x-a)#.

So the simplest polynomial function with the required properties is:

#f(x) = 2(x-0)(x-3-4i)(x-3+4i)#

#color(white)(f(x)) = 2x((x-3)^2-(4i)^2)#

#color(white)(f(x)) = 2x(x^2-6x+9+16)#

#color(white)(f(x)) = 2x(x^2-6x+25)#

#color(white)(f(x)) = 2x^3-12x^2+50x#