Given the word problem, we can deduce the following information:
The zeroes are: 4, 3, 8
To determine the polynomial function P(x) with the given zeroes, we follow the process as shown below:
[tex]\begin{gathered} (x-4)(x-3)(x-8)=0 \\ \\ \end{gathered}[/tex]We first expand (x-4)(x-3):
[tex](x-4)(x-3)=x^2-7x+12[/tex]Next, we expand (x^2-7x+12)(x-8):
[tex]\begin{gathered} (x^2-7x+12)(x-8)=x^2(x)+x^2(-8)-7x(x)-7x(-8)+12(x)+12(-8) \\ Simplify \\ =x^3-15x^2+68x-96 \end{gathered}[/tex]Hence,
[tex]x^3-15x^2+68x-96=0[/tex]Therefore, the polynomial function is:
[tex]P(x)=x^3-15x^2+68x-96[/tex]