Question

If r is a complex number and p(z) is a non-constant polynomial with complex coefficients, then there exists a polynomial q(z) and a constant c such that:

a) p(z) = q(z) + c
b) p(z) = c * q(z)
c) p(z) = q(z) * c
d) p(z) = q(z) + r

Answer 1

A non-constant polynomial p(z) with a complex number r can be represented as another polynomial q(z) when p(z) = q(z) + r, by defining q(z) = p(z) - r.

Step-by-step explanation:

If r is a complex number and p(z) is a non-constant polynomial with complex coefficients, we can create another polynomial q(z) such that p(z) = q(z) + r. This is simply done by setting q(z) = p(z) - r. Since r is a constant complex number, subtracting it from p(z) leaves us with another polynomial q(z) which when r is added back to it, gives us the original polynomial p(z).