Question

Find a polynomial with integer coefficients, with leading coefficient 1, degree 5, zeros i and 8-i, and passing through the origin

Answer 1

the required polynomial is a product of the linear and/or quadratic factors with real and/or complex roots or zeroes.

one zero is i, therefore its conjugate -i is also a zero.

the resulting quadratic equation with zeroes i and -1 is x^2+1=0, and the quadratic factor needed is x^2 + 1

another zero is (8-i), with its conjugate (8+i) also a zero. the resulting quadratic factor with integer coefficients is \[\large [x-(8-i)][x-(8+i)]\]

since the required degree is 5, there is an unknown linear factor of the form \(\large (x-r)\)

the required polynomial is \[\large p(x)=(x-r)(x^2+1)(x^2-16x+65)\]

to find r, use the last hint, "passes through the origin", that is, x=0 and p(x)=0