Final answer:
The polynomial p(x) of least degree with rational coefficients, a leading coefficient of one, and roots -2, 3, and 6 is found by creating factors from each root, resulting in p(x) = (x + 2)(x - 3)(x - 6), which expands to x^3 - 6x^2 - x + 36.
Step-by-step explanation:
The polynomial p(x) of least degree with a leading coefficient of one and with roots -2, 3, and 6 can be found by using the roots to create factors of the polynomial. Each root 'r' of the polynomial with rational coefficients gives rise to a factor of the form (x - r). Given the roots -2, 3, and 6, the factors corresponding to these roots would be (x + 2), (x - 3), and (x - 6), respectively.
Combining these factors, we get:
p(x) = (x + 2)(x - 3)(x - 6)
We can then expand these factors to find the polynomial:
p(x) = x3 - 6x2 - x + 36
This polynomial satisfies the conditions: it has rational coefficients, a leading coefficient of one, and the given roots.