Convolution and Polynomial Multiplication

Suppose we have two signals f(n) and g(n). Convolution is Mathematically defined as:

Let f(n) is as shown in the following figure:
And let g(n) is shown bellow:
Computing the convolution f(n) * g(n), we have the following results….

Now consider two polynomials:and

Note the relationship of the signal f(n) with the polynomial f. Similarly, note relationship of the signal g(n) with the polynomial g.
Multiplying these two polynomials we get:

Try taking different signals f(n) and g(n) and their corresponding polynomials f and g. Compute f(n) * g(n) and f X g.
Can we draw any conclusions with the results?

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