Convolution vs multiplication vs polynomial multiplication?
The convolution theorem is a fundamental signal processing property that links the frequency and time domains. Specifically convolution in one domain is multiplication in the other. Often this can be used to simplify the number of operations required for solving signal processing problems. The utility of this is amplified given the significance of the Fast Fourier Transform in how efficiently it can be used to translate between domains. 

With this in mind, consider that the output of a linear system in time is related to its input using convolution:

$$y(t) = x(t)\star h(t)$$

Where $\star$ represents the convolution operator, $x(t)$ is the input and $h(t)$ is the impulse response of the system.

If we took the Fourier Transform of $x(t)$ and $h(t)$, then this same problem is solved using an element by element product:

$$Y(\omega) = X(\omega)H(\omega)$$   

Similarly we use convolution to multiply polynomials by convolving the coefficients. So we could use the FFT for polynomial multiplication as follows:

Consider the simple example of a product of $x^3 + 2x^2 - 4x +1$ with $5x^2+3$

This is solved by convolving the coefficients as:

$$[1, 2, -4, 1] \star [5, 0, 3] = [5, 10, -17, 11, -12, 3]$$

Resulting in $5x^5 + 10x^4 -17x^3+11x^2-12x+3$

Similarly this could be done in the frequency domain with FFT processing, but to do this properly the coefficients of the polynomial are loaded starting with the lowest power of x and we must zero-pad out to 6 samples that we would expect in the result:

FFT of $[1,-4, 2,1,0,0] = [0, -3+1.73j, 3+5.20j, 6, 3-5.2j, -3-1.73j]$

FFT of $[3,0,5,0,0,0] = [8, 0.5-4.33j, 0.5+4.33j, 8, 0.5-4.33j, 0.5+4.33j]$

Element by element product of the FFTs: $[0, 6+13.85j, -21+15.59j, 48, -21-15.59j, 6-13.95j]$

Inverse FFT of the product:  $[3, -12, 11, -17, 10, 5]$

Corresponding to $5x^5+10x^4-17x^3+11x^2-12x+3$

For something this small as in this example, it would be more efficient to do the convolution directly, but as the length of the polynomials grow, the FFT approach becomes significantly more efficient.