Does multiplying polynomials ever decrease the number of terms?

Let p and q be polynomials (maybe in several variables, over a field), and suppose they have m and n non-zero terms respectively. We can assume mn. Can it ever happen that the product pq has fewer than m non-zero terms?

I ask this because I vaguely recall seeing a positive answer in book somewhere (probably about computation or algorithms since the polynomials were unwieldy). If anyone knows what book this is from it would be much appreciated.

Answer

(x22x+2)(x2+2x+2)=x4+4.

Attribution
Source : Link , Question Author : Chris Brooks , Answer Author : André Nicolas

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