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For the sake of simplicity, in the sequel a (Laurent) polynomial of negative degree is understood to be the zero polynomial and, similarly, a banded matrix with negative bandwidth reduces to the zero matrix.
As shown below, the roots of a polynomial are the values of x that make the polynomial zero, so they are where the graph crosses the x-axis, since this is where the y value (the result of the polynomial) is zero. The roots are the two green dots.