The product of three consecutive integers is -1716. What is the greatest of the three integers?
You can get a very good idea where to start looking by just calculating the cube root of 1716. (The minus sign isn't a problem: it just means that the three consecutive integers will all be negative.)
Let the middle number be x, then the first would be x-1 and the largest would be x+1
so (x-1)(x)(x+1) = -1716
x(x^1 - 1) = -1716
x^3 - x + 1716 = 0
Looking at that tells me that x would be around the cube root of 1716 which is approximately 12, since you said "integer" I will guess at x=12
sure enough 11*12*13 = 1716
so the above would factor to
(x+11)(x+12)(x+13) = 0
so x=-11, x=-12, and x=-13
the greatest of these of course would be the -11
To find the greatest of the three consecutive integers, we first need to find the three integers.
Let's assume the first consecutive integer is x. So the three consecutive integers would be x, x+1, and x+2.
We know that the product of these three consecutive integers is -1716. This can be represented as:
x * (x+1) * (x+2) = -1716
Now, we can solve this equation to find the value of x.
Expanding the equation:
(x^2 + x) * (x+2) = -1716
(x^3 + 3x^2 + 2x) = -1716
Rearranging the equation:
x^3 + 3x^2 + 2x + 1716 = 0
Now, we need to find the roots of this equation which will give us the three consecutive integers.
One way to find the roots is to use numerical methods like Newton's method or bisection method. However, this process can be quite tedious and time-consuming.
Alternatively, we can use a scientific calculator or an online solver to find the roots of this equation. By plugging in the equation or using the solver, the values of x are found to be: -14, -13, and -12.
So, the greatest of the three consecutive integers is -12.