How to find that the given equation has integer roots?

Question

How we will find that given equation ( either linear , quadratic, biquadratic, etc) have integer roots ? And also is there any way to find the number of integer roots?

Answer 1

When the equation is in one variable, you can use the rational root theorem like Peter suggested: if some $n\in\mathbb Z$ satisfies $p(n)=0$, then $n\mid p(0)$. (I'm assuming $p$ is monic here.) But so long as you go beyond one variable, this problem becomes very complicated and there is definitely no general way of determining this. In fact, in some sense, finding whether integer solutions exist to integer equations in more than one variable is exactly the central question algebraic geometry tries to answer. Algebraic geometry is still a very active field of research.

For example, equations of the form $y^2=x^3+ax^2+bx+c$ are often known as ellitpic curves, and it still does not exist a general method of finding whether this has integer solutions. Indeed, if you can answer this, then by the other established results in the study of ellitpic curves, we would have a near complete theory. You'd probably earn a big prize for this.

Yet another example: Fermat's equation $x^n+y^n=z^n$ where $n\geq3$. Needless to say, finding whether this has (nontrivial) integer solutions is equivalent to answering Fermat's Last Theorem, which is obviously not an easy task. Wiles had resolved this problem, proving that no nontrivial integer solutions exist, but his proof builds upon decades of prior work and is something that would take an immense amount of work to comprehend.

In conclusion: when there is only one variable, there's only finitely many cases to check, which is easy. Once you get beyond that, no established methods exist.

Answer 2

If you have a polynomial with integer coefficients, the possible integer roots are the divisors (also the negative divisors) of the constant term.

In the case , the constant term is zero, $0$ is an integer root. In this case, factor out the highest possible power to reduce the problem to the case where the constant term is non-zero.