Final answer:
The polynomial p(x) in factored form with the given requirements is p(x) = (x - 5)(x - 3)(x + 2)(x - 4).
Step-by-step explanation:
To write a polynomial p(x) in factored form with a degree of 4, a leading coefficient of 1, x-intercepts at (5,0), (3,0), (-2,0), and (4,0), and a y-intercept at (0, -120), we use the given x-intercepts to create factors for p(x). Each x-intercept represents a root of the polynomial, so the factors are:
- (x - 5)
- (x - 3)
- (x + 2)
- (x - 4)
The polynomial in factored form is then p(x) = (x - 5)(x - 3)(x + 2)(x - 4). To ensure the proper y-intercept, we can multiply our polynomial by a constant a such that p(0) = a*(0 - 5)*(0 - 3)*(0 + 2)*(0 - 4) = -120. Solving for a gives us a = 1, which confirms that the leading coefficient is 1 and there's no need for further adjustment to match the y-intercept.
Therefore, the polynomial in factored form is p(x) = (x - 5)(x - 3)(x + 2)(x - 4).