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
Explanation:
If a polynomial has rational coefficients, then any radical or complex zeros will occur in conjugate pairs.
So if
A polynomial has a zero
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#