ynomial Functions
Question 12, 2.5 .31
HW Score: $43.43 \%, 14.33$ of 33 points
Points: 0 of 1
Sav
Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value.
\[
\begin{array}{l}
n=4 ; \\
-1,3 \text {, and } 4+3 i \text { are zeros; } \\
f(1)=-144
\end{array}
\]
\[
f(x)=[
\]
(Type an expression using $x$ as the variable. Simplify your answer.)

Step 1 ：Given that the polynomial function has real coefficients, if a complex number is a root, then its conjugate is also a root. Therefore, the other root is \(4 - 3i\).

Step 2 ：The polynomial function is given by: \[f(x) = a(x - (-1))(x - 3)(x - (4 + 3i))(x - (4 - 3i))\]

Step 3 ：Simplify the expression: \[f(x) = a(x + 1)(x - 3)(x^2 - 8x + 25)\]

Step 4 ：We know that \(f(1) = -144\), so we can substitute \(x = 1\) into the equation to solve for \(a\): \[-144 = a(1 + 1)(1 - 3)(1 - 8 + 25)\]

Step 5 ：Solving the equation gives: \[-144 = a(2)(-2)(18)\] \[-144 = -72a\] \[a = 2\]

Step 6 ：So, the polynomial function is: \[f(x) = 2(x + 1)(x - 3)(x^2 - 8x + 25)\]

Step 7 ：Simplified, this becomes: \[f(x) = 2x^4 - 30x^2 + 144x - 144\]

Step 8 ：This is the simplest form of the polynomial function. The final answer is \[\boxed{f(x) = 2x^4 - 30x^2 + 144x - 144}\]