Question: Construct a polynomial function with the following properties: fifth degree, 2 is a zero of…

Construct a polynomial function with the following properties: fifth degree, 2 is a zero of multiplicity 3, -4 is the only other zero, leading coefficient is 5.

Answer:

\[ f(x) = \]

Solution

Let’s construct the polynomial step by step. First, identify the zeros and their multiplicities. \[ \begin{cases} 2 \text{ with multiplicity } 3 & \\ -4 \text{ with multiplicity } 3 & \end{cases} \] Next, write the factors corresponding to each zero. \[ (x - 2)^3 (x + 4)^3 \] Finally, include the leading coefficient to form the polynomial. \[ f(x) = 5(x - 2)^3 (x + 4)^3 \]