Answer:
To write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros -2, 3, 6, we can use the Linear Factorization Theorem1.
The theorem states that if a polynomial function f has zeros x = r1, x = r2, …, x = rn (where r1, r2, …, rn are distinct real numbers), then f(x) can be factored as follows:
f(x) = a(x - r1)(x - r2)…(x - rn)
where a is a nonzero constant.
So for this problem, we have:
f(x) = (x - (-2))(x - 3)(x - 6)
Multiplying this out gives:
f(x) = (x + 2)(x - 3)(x - 6)
Expanding this expression gives:
f(x) = x^3 - 7x^2 + 12x + 36
Therefore, the polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros -2, 3, 6 is:
f(x) = x^3 - 7x^2 + 12x + 36
I hope this helps! Let me know if you have any other questions.
Explanation: