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No, it is not. f(x) = 2x + 3 and g(x) = 3x2 are polynomials but f(x)/g(x) is not a polynomial.
(x - 3)(x - 3)
(x + 4)(x - 3)
Not quite. The polynomial's linear factors are related - not equal to - the places where the graph meets the x-axis. For example, the polynomial x2 - 5x + 6, in factored form, is (x - 2) (x - 3). In this case, +2 and +3 are "zeroes" of the polynomial, i.e., the graph crosses the x-axis. That is, in an x-y graph, y = 0.
(x-3)(x+8)