Question

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?

Answer 1

`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`