Final answer:
The polynomial P(x) of degree 5 with leading coefficient 1 and zeros at (1,0), (−9,0), (2,0), and (−2,0), and a y-intercept at (0,−180) is P(x) = (x - 1)(x + 9)(x - 2)(x + 2)(x + 5).
Step-by-step explanation:
To write a polynomial, P(x), of degree 5 with a leading coefficient of 1 and given zeros, we can express it in factored form. The zeros of the polynomial are given at (1,0), (−9,0), (2,0), and (−2,0). Since we need a polynomial of degree 5 and we have only 4 distinct zeros, one of the zeros must be repeated. The y-intercept at (0,−180) will be used to determine the constant coefficient.
The factored form of the polynomial with the given zeros will be:
P(x) = (x - 1)(x + 9)(x - 2)(x + 2)(x - c)
Here, 'c' represents the zero that we need to determine. Since the leading coefficient is 1, we do not multiply by any other coefficient. To find 'c', we use the fact that the y-intercept of the polynomial is (0,−180). By substituting x = 0 into P(x), we can solve for 'c'.
When x=0, P(0) = (0 - 1)(0 + 9)(0 - 2)(0 + 2)(0 - c) = -180
This simplifies to (-1)(9)(-2)(2)(-c) = -180
Calculating the product of the numbers on the left, we get 36c = -180, which, when solved, gives c = -5.
The polynomial in factored form with the determined value of 'c' is:
P(x) = (x - 1)(x + 9)(x - 2)(x + 2)(x + 5)