Answer:
p(x) = -x⁵ -4x⁴ +9x³ +40x² +4x -48
Explanation:
You want a polynomial of 5th degree with x-intercept (3, 0), point (-2, 0) a local maximum, and a limit as x→-∞ of p(x)→∞.
Determined characteristics
The requirements placed on the polynomial mean that the graph will cross the x-axis at x=3 and will touch the x-axis from below at x=-2. The touch from below means the zero at x=-2 must have even multiplicity. That multiplicity can only be 2 in order for the other requirements to be met.
Since the polynomial is positive for large negative x-values, the leading coefficient must be negative, and there must be a zero that is less than -2.
So far, we have four (4) zeros accounted for, but we need 5. The 5th zero must be greater than -2.
Electives
For the polynomial shown in the graph, we have elected to have zeros at x=-4 and x=1. Each zero at x=q means (x-q) is a factor. The multiplicity of the zero is the exponent of the factor.
This makes the factored form be ...
p(x) = -(x +4)(x +2)²(x -1)(x -3)
The expanded form is ...
p(x) = -x⁵ -4x⁴ +9x³ +40x² +4x -48