Join the points using a smooth curve. If you have n points choose a polynomial of degree at most (n-1). You will always be able to find polynomials of degree n or higher that will fit but disregard them.
The roots are the points at which the graph intersects the x-axis.
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Descartes' rule of signs (see related link) can help you determine the maximum number of real roots. If the polynomial is odd powered, then there will be at least one real root. Any even powered polynomial can be factored into a bunch of quadratics [though they may not be rational or even pretty], and any odd-powered polynomial can be factored into a bunch of quadratics and one linear (this one would have the real root). So the quadratics may have pairs of real or complex roots (having an imaginary component).To clarify, when I say complex, I'm referring to the fact that there will be an imaginary component to the root, because actually the real numbers is a subset of the set of complex numbers.The order of the polynomial will tell you how many roots it will have. If you can graph the polynomial, then you can see if it crosses the x axis. If it is a 5th order polynomial, and crosses the x axis 3 times, then there are 3 real roots (the other two roots are complex).
13 is not a polynomial.
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A single polynomial cannot have a greatest commonfactor. There is nothing that it will be in common with!
Rational zero test cannot be used to find irrational roots as well as rational roots.