# Finding an integer solutions of a polynomial function

Integer solutions of a polynomial function theorem says: If a polynomial function with integer coefficients has an integer solution, , then that solution is the divisor of free coefficient .

As an addition to this theorem, for every whole number k, number is a divisor of .

Example 1. Find all integer roots of .

The set of dividers is, according to the last theorem, the set of dividers of number -26.

It would take a lot of time to check all these solutions so we’ll first eliminate few of them.

We can take any whole number k, for example here we’ll take k = 1. According to the addition to the theorem, for every integer root of this polynomial – , polynomial has to divide f(k) without the remainder.

Now let’s take a look at a set of solutions for .

Now we can eliminate every with which – 8 is not divisible.

{-2, 0, -3, 1, -14, 12, -27, 25}

Now we have significantly smaller set of possible solutions. Now we have to check both of them to see are they roots of given equation.

This means that number 2 is the only whole root of this equation.

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