The product of three consecutive integers is - 1716. what is the greatest of the three integers?

-11

Step-by-step explanation:

If we call x the least integer, then x + 1 is the next integer, and x + 2 is the greatest integer.

x (x + 1) (x + 2) = - 1716

x (x² + 3x + 2) = - 1716

x³ + 3x² + 2x = - 1716

x³ + 3x² + 2x + 1716 = 0

That's not an easy equation to factor. We could use rational root test, but that would take a long time. However, we can narrow down the options logically. The three integers are relatively close together, so if we approximated the product as x³:

x³ ≈ - 1716

x ≈ - 11.9

So let's see if x = - 12 is a solution:

(-12) ³ + 3 (-12) ² + 2 (-12) + 1716 = 396

Nope. Let's try x = - 11:

(-11) ³ + 3 (-11) ² + 2 (-11) + 1716 = 726

Not that either. Let's go the other direction, x = - 13:

(-13) ³ + 3 (-13) ² + 2 (-13) + 1716 = 0

Aha! x = - 13. So x + 1 = - 12, and x + 2 = - 11.

-11 * - 12 * - 13 = - 1716

The greatest integer is - 11.