How do you write a polynomial function of least degree with integral coefficients that has the given zeroes 1 + 2i, 1- i?
1 Answer
Mar 11, 2016
Explanation:
The first notion is that a polynomial with zeroes that have imaginary components must have pairs of conjugate complex roots so that the polynomial has real coefficients
Then the polynomial of the problem must have these roots
So the required polynomial is
#(x-1+2i)(x-1-2i)(x-1+i)(x-1-i)#
#(x^2-2x+1+4)(x^2-2x+1+1)#
#(x^2-2x+5)(x^2-2x+2)#
#x^4-2x^3+2x^2-2x^3+4x^2-4x+5x^2-10x+10#
#-> x^4-4x^3+11x^2-14x+10#
