To find a polynomial function \( f(x) \) of the least degree with the given zeros, follow these steps:
1. Identify the given zeros:
- Zero at \( x = 3 \) with multiplicity 2.
- Zero at \( x = 4i \). Since the polynomial must have real coefficients, the complex conjugate \( x = -4i \) is also a zero.
2. Write the factors corresponding to each zero:
- For the zero at \( x = 3 \) with multiplicity 2: \((x - 3)^2\).
- For the zero at \( x = 4i \): \((x - 4i)\).
- For the zero at \( x = -4i \): \((x + 4i)\).
3. Combine all factors to form the polynomial:
Combine the factors to construct the polynomial:
[tex]\[
f(x) = (x - 3)^2 (x - 4i) (x + 4i)
\][/tex]
4. Simplify the polynomial:
- First expand the complex conjugate pair product:
[tex]\[
(x - 4i)(x + 4i) = x^2 - (4i)^2 = x^2 - (-16) = x^2 + 16
\][/tex]
- The polynomial now is:
[tex]\[
f(x) = (x - 3)^2 (x^2 + 16)
\][/tex]
5. Expand the remaining factor:
- Expand \((x - 3)^2\):
[tex]\[
(x - 3)^2 = x^2 - 6x + 9
\][/tex]
- Now multiply this with \(x^2 + 16\):
[tex]\[
f(x) = (x^2 - 6x + 9)(x^2 + 16)
\][/tex]
6. Perform the final expansion:
- Multiply each term in the first polynomial by each term in the second polynomial:
[tex]\[
\begin{align*}
f(x) &= x^2(x^2 + 16) - 6x(x^2 + 16) + 9(x^2 + 16) \\
&= x^4 + 16x^2 - 6x^3 - 96x + 9x^2 + 144 \\
&= x^4 - 6x^3 + 25x^2 - 96x + 144
\end{align*}
\][/tex]
So, the polynomial of the least degree with the given properties is:
[tex]\[
f(x) = x^4 - 6x^3 + 25x^2 - 96x + 144
\][/tex]
