A degree 4 polynomial with integer coefficients has zeros 14

A degree 4 polynomial with integer coefficients has zeros 14i and 1, with 1 a zero of multiplicity 2. If the coefficient of x4 is 1, then the polynomial is

Solution

zeros 14i and 1, with 1 a zero of multiplicity 2

f(x) has another zero which is complex conjugate of -1-4i = -1+4i

f(x) has zero at x=1

f(x) has zero at x =2

So, f(x) = k.(x-1)(x-2)(x+1+4i)(x+1-4i)

where k iscoefficient of x^4 is which is 1.So, k=1

f(x) = 1.(x^2 -3x +2)( x^2 +x -4ix +x+1 -4i +4ix+4i +16)

=(x^2 -3x+2)(x^2 +2x+17)

=( x^4 +2x^3 +17x^2 -3x^3 - 6x^2 -51x +2x^2 +4x + 34)

f(x) = ( x^4 -x^3+13x^2 -47x +34)