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A polynomial in x has m nonzero terms. Another polynomial in x has n nonzero terms, where m is less than n. These polynomials are multiplied and all like terms are combined. The resulting polynomial has a maximum of how many nonzero terms? How would you prove that the answer is mn?

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\begin{align} & (A+B+C+D)(X+Y+Z) \\[12pt] = {} & \phantom{{}+{}} A(X+Y+Z) \\ & {} + B(X+Y+Z) \\ & {} + C(X+Y+Z) \\ & {} + D(X+Y+Z) \\[12pt] = {} & \phantom{{}+{}} AX+AY+AZ \\ & {} + BX+BY+BZ \\ & {} + CX+CY+CZ \\ & {} + DX+DY+DZ \\[12pt] = {} & \text{a sum of }4\times3\text{ terms}. \end{align}

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Each non-zero coefficient in the product must be the result of at least one pair from the original polynomials so the number cannot exceed $mn$.

$mn$ is a possible result, for example from $(x^{(m-1)n}+x^{(m-2)n}+\cdots+x^{2n}+x^{n}+1)(x^{n-1}+x^{n-2}+\cdots+x^2+x^1+1)$

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