Question

Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree​ 4; ​ zeros: 5-4i multiplicity 2 Let a represent the leading coefficient. The polynomial is f(x) = a()

Answer 1

Given the zeros \(5 - 4i\) with multiplicity 2, the conjugate \(5 + 4i\) is also a zero due to the complex conjugate root theorem.

So, the factors of the polynomial are \((x - (5 - 4i))^2\) and \((x - (5 + 4i))^2\).

Now, expand these factors:

\[ (x - (5 - 4i))^2 = (x - 5 + 4i)^2 \]
\[ (x - (5 + 4i))^2 = (x - 5 - 4i)^2 \]

Multiply these to get the polynomial:

\[ f(x) = a(x - 5 + 4i)^2(x - 5 - 4i)^2 \]

The leading coefficient 'a' represents the coefficient of the highest-degree term, but since it's not specified, you can leave it as \( a(x - 5 + 4i)^2(x - 5 - 4i)^2 \).