What is the third–degree polynomial function such that f(0) = –18 and whose zeros are 1, 2, and 3

The zeroes are 1, 2, and 3, so the factors are (x-1), (x-2), and (x-3). Those are 3 factors for a 3rd degree polynomial, so no other factors are needed.

f(x) = C*(x-1)(x-2)(x-3), where C is any constant.

f(0) = 18 allows you to determine C.

To find the third-degree polynomial function, we need to use the given zeros and the point (0, -18).

A polynomial function with zeros at 1, 2, and 3 will have factors of (x-1), (x-2), and (x-3). If we multiply these factors together, we will get the polynomial function.

So, the general form of the third-degree polynomial function is:

f(x) = a(x-1)(x-2)(x-3)

To find the value of 'a', we can plug in the point (0, -18) into the function:

-18 = a(0 - 1)(0 - 2)(0 - 3)

Simplifying this equation:

-18 = a(-1)(-2)(-3)
-18 = -6a

Dividing both sides by -6:

-18 / -6 = a
3 = a

Now, we know that 'a' is equal to 3. So, the third-degree polynomial function is:

f(x) = 3(x-1)(x-2)(x-3)

Note: The leading coefficient of the polynomial determines the shape of the graph. In this case, since the leading coefficient is positive, the graph will open upward.