# Solving equations using the rational root theorem

We want to solve the equation x^3 - 44 x^2 - 516 x + 10800 =0 using the rational root theorem. If an algebraic equation with integer coefficients has a rational solution then the numerator of the solution must divide the constant term, which is 10800 in this case and the denominator must divide the coefficient to the term with the highest power of x, which is 1. This means that any rational solution will also be an integer solution. The problem we're then faced with is that 10800 has a large number of divisors. Calculate how many divisors 10800 has by factoring this number and then try to find a workaround way to get around the problem of having to try out all solutions.

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#### Solution Preview

The constant term is We have 10800 = 2^4*3^3 5^2. This has divisors of the form s * 2^a * 3^b * 5^c where s is a sign, and a is between 0 and 4, b is between 0 and 3 and c is between 0 and 2. This means that there are 2*5*4*3 = 120 possible divisors to check out.

We can get around the problem of checking out all these number using a method mentioned in [1]. It's convenient to define the function

f(x) = x^3 - 44 x^2 - 516 x + 10800

The function g(y) = f(y+p) for some arbitrary integer p is also a third degree polynomial ...

#### Solution Summary

I explain how to apply the rational root theorem when there are a very large number of possible solutions.