How do you create a polynomial with Degree 4, 2 Positive real zeros, 0 negative real zeros, 2 complex zeros?

1 Answer
Nov 26, 2015

Answer:

Any polynomial which factors to the form
#c(x-a)(x-b)(x-di)(x+di)# with #a, b, c, d in RR^+#
will satisfy the given conditions.

Explanation:

A simple way to creative a polynomial with certain desired zeros is to write it first in its factored form. For example, in this case, let's make a polynomial with the zeros #1, 2, i, -i#.

Note that the solutions to
#(x-1)(x-2)(x-i)(x+i)= 0#
are #1, 2, i, -i#

Thus we know that #(x-1)(x-2)(x-i)(x+i)# is a polynomial with the desired properties. Then, all that remains is to multiply it out.

#(x-1)(x-2)(x-i)(x+i) = (x-1)(x-2)(x^2+1)#

#= (x-1)(x^3 -2x^2 + x -2)#

#=x^4 -3x^3 +3x^2 -3x +2#

While the above works, #1, 2, i, -i# were chosen arbitrarily. Any polynomial which factors to the form
#c(x-a)(x-b)(x-di)(x+di)# with #a, b, c, d in RR^+# will satisfy the given conditions.