posted by Kyle .
Find a polynomial with integer coefficients that satisfies the given conditions.
R has degree 4 and zeros 3 − 4i and 5, with 5 a zero of multiplicity 2.
x = 3 - 4i
x = 3 + 4i
x = 5
two 5's because multiplicity 2
You always have to have the conjugate of an imaginary number.
(x -(3-4i))(x-(3+4i)(x-5)(x-5) = 0
Easiest way is to multiply the first two and the second two factors.
Then multiply those answers together to get the polynomial.
(x-5)(x-5)(x-3+4i)(x-3-4i) = 0
multiply that out
complex values come in pairs (why?), so the roots are
3-4i, 3+4i, 5, 5
R(x) = (x-(3-4i))(x-(3+4i))(x-5)^2
= x^4 - 16x^3 + 110x^2 - 400x + 625