The volume of a box is modeled by the function V(x) = x3 + 2x2 − 29x − 30. Identify the values of x for which the volume is 0 and...

We are given the model for the volume of a box as  we
and we are asked to factor the function:

If the function factors in the rationals, we can write the function as
V(x)=(x-p)(x-q)(x-r), where p,q, and r are the real zeros of the function.

If we have access to a graphing utility, we can graph the function and use the zeros to factor; here is the graph:

The zeros appear to be at x=-6, x=-1, and x=5. If this is true, the function factors as:

V(x)=(x+6)(x+1)(x-5)

------------------------------------------------------------------------------

The zeros are at x=-6,-1, and 5. The factored form is:

V(x)=(x+6)(x+1)(x-5)

-----------------------------------------------------------------------------

If we did not have access to a grapher, by the rational root theorem we know that the only possible rational roots are

It would not take long to find the roots.