Question

if a polynomial function p(x) has zeros, 5i,0,and 3 of multiplicity two, which of the following represents p(x)

Answer 1

Im going assume we are talking about polynomials with real coefficients here.

For every zero of the function, say r, there is a term:\[(x-r)\]inside of the polynomial. If it has a multiplicity of k, then make that the power:\[(x-r)^k\]So for the zeros 0 and 3 (multiplicity 2), we end up with:\[(x-0)(x-3)^2=x(x-3)^2\]All that is left to deal with is the last zero 5i, but this is a complex number. If a complex number is a zero of a real polynomial, then its complex conjugate must also be a root. The complex conjugate of 5i is -5i, so we have:\[(x-5i)(x-(-5i))=(x-5i)(x+5i)\]Altogether now:\[x(x-3)^2(x-5i)(x+5i)\]I leave it to you to multiply everything out.