Mark M. answered • 02/18/16

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Since f(x) is a polynomial function, it is continuous for all x.

The Intermediate Value Theorem says that if f(x) is continuous on an interval [a,b] and if f(a)and f(b) have opposite signs, then f(x) = 0 for some number in the interval (a,b).

Geometrically, if the graph of a continuous function lies below the x-axis at one endpoint of an interval and lies above the x-axis at the other endpoint of the interval, then the graph must cross the x-axis at least once between the interval's endpoints.

For the given continuous function, f(3) = -2 < 0 and f(4) = 14 > 0. So, by the Intermediate Value Theorem, f(c) = 0 for some number c in the interval (3,4).