Find a polynomial with integer coefficients with leading coe

Find a polynomial with integer coefficients, with leading coefficient 1, degree 5, zeros i and 8-i, and passing through the origin. P(x)=

Solution

P(x) passes through origin , so x=0 is a zero

Given zeros are i and 8-i,

Complex conjugate zero of i = -i

Complex conjugate zero of 8+i

P(x) = a*x(x^2 +1)( x - 8 +i)( x- 8 -i)

Leading coeffeicient 1 so, a=1

P(x) = x(x^2 +1)( x - 8 +i)( x- 8 -i)

= x(x^2 +1)( x^2 -8x -ix -8x +64 +8i +ix -8i +1)

P(x) = x( x^2 +1)( x^2 -16x +1) ( Solution)