Find a polynomial function with the zeros -2,3,5 whose graph passes through the point (6,72).

1. Given the zeros are -2, 3, 5 so the polynomial in factored form is:

P(x) = a(x + 2)(x - 3)(x - 5)

2. Use the point (6, 72) to find a :

72 = a(6 + 2)(6 - 3)(6 - 5)

3. Simplify inside the parentheses:

72 = a(8)(3)(1)

4. Solve for a :

72=24\Rightarrow a=3

5. Substitute a = 3 back into the polynomial:

P(x) = 3(x + 2)(x - 3)(x - 5)

6. Expand the polynomial:

3[(x^2 - x - 6)(x - 5)] = 3(x^3 - 6x^2 - x + 30)

7. Final polynomial function is:

P(x) = 3x^3 - 18x^2 - 3x + 90

8. Answer:

\boxed{P(x) = 3x^3 - 18x^2 - 3x + 90}