Can the Product of Two Polynomials Result in a Single Term?

     By assuming that we are talking about polynomials with real coefficients. Suppose that the polynomials are, 

P(x) = axp +  ...  +  bxq
Q(x) = cxr +  ...  +  dxs
Where . . .
  • a, b, c, d 0
  • p > q 0 and r > s ≥ 0
  • p + r > q + s
  • Of course, the terms omitted between have intermediate degree.
Then, the product  PQ has at least two terms: acxp+r and bdxq+s. Where ac and bd are not zero.

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