Question

Form a polynomial, f(x), with real coefficients having the given degree and zeros. Degree: 4; zeros: 3+2i; 4, multiplicity 2

Answer 1

\[
x = 3 + 2i \\
x - 3 = 2i \\
(x-3)^2 = (2i)^2 = -4 \\
x^2 - 6x + 9 = - 4 \\
x^2 - 6x + 13 = 0 \\
\text{ } \\
x = 4 \\
(x-4) = 0 \\
\text{multiplicity of 2 makes it:} (x-4)^2 \\
f(x) = (x^2 - 6x + 13)(x-4)^2 \\
\text{Multiply it out and arrange it in decreasing powers of x}
\]