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In a polynomial function, the variable x is raised to some integer power.

f(x) = 5x³ + 8x⁵

g(x) = (x + 5)²

In an exponential function, some real number is raised to the power of variable x or some function of x

f(x) = 5ˣ

g(x) = eˣ⁺²

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Q: What is the difference be a polynomial function and an exponential function?
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Example of fundamental difference between a polynomial function and an exponential function?

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Do you mean, "the difference between an algorithm that runs in polynomial time, and one that runs in exponential time".First a real quick review. A polynomial is any equation of the formy = cmxm + ... + c2x2 + c1x + c0 ,where ci are constantsAn exponential function is something of the formy = cxThese functions grow much faster than any polynomial function.So, if T(n) describes the runtime of an algorithm as a function of whatever (# of inputs, size of input, etc.)., and T(n) can be bound above by any polynomic function, then we say that algorithm runs in polynomial time.If it can't be bound above by a polynomial function, but can be bound above by an exponential function, we say it runs in exponential time.Note how ugly an exponential algorithm is. By adding one more input, we roughly double (or triple, whatever c is) the run-time.


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