Final answer:
The polynomial function p(x) with leading coefficient 1, least possible degree, and zeros -8 and 3 is p(x) = x^2 + 5x - 24.
Step-by-step explanation:
To find the polynomial function p(x) having leading coefficient 1, least possible degree, and the given zeros, -8 and 3, we use the fact that the zeros of a polynomial function are the values of x that make the function equal to zero. Since a polynomial with real coefficients will have factors corresponding to each zero, and the polynomial should have the least possible degree, we can construct it by creating factors associated with each zero and then expanding these factors. The factors related to the zeros -8 and 3 are (x + 8) and (x - 3), respectively. Because the leading coefficient must be 1, we do not need to multiply these factors by any other coefficients. The polynomial function is found by multiplying these factors together:
p(x) = (x + 8)(x - 3)
Expanding the factors, we get:
p(x) = x^2 - 3x + 8x - 24
p(x) = x^2 + 5x - 24
Therefore, the polynomial function p(x) with a leading coefficient of 1, least possible degree, and zeros -8 and 3 is p(x) = x^2 + 5x - 24.