Find the polynomial g(x) with integer coefficients of least degree (and positive leading coefficients as small as possible) for which 1-2i is a double root?

1 Answer
Oct 5, 2017

#g(x)=x^4-4x^3+14x^2-20x+25#

Explanation:

As the polynomial #g(x)# has integer coefficients and #1-2i# is a double root of the polynomial, it has its factor #(x-(1-2i))^2#.

But it will give us complex coefficients and hence for integers as coefficients we must have another double root which is complex conjugate of #1-2i# i.e. #1+2i#. Hence #(x-(1+2i))^2# too is a factor.

Hence the polynomial with least degree would be

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

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

= #(x^2-xcolor(red)(-2ix)-x+1color(red)(+2i+2ix-2i)-4i^2)^2#

= #(x^2-2x+1+4)^2#

= #(x^2-2x+5)^2#

= #x^4+4x^2+25-4x^3+10x^2-20x#

(using #(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2ab#)

= #x^4-4x^3+14x^2-20x+25#