If f(x) is a polynomial of degree 7, and g(x) is a polynomial of degree 7, then what is the product of the minimum and the maximum possible degrees of f(x) + g(x)?

 May 22, 2020

If the two coefficients of the highest order terms of both polynomials have the same sign, then the sum is also a 7th degree polynomial.

If the highest order terms have coefficients that are of equal magnitude but opposite sign, then the sum would be a polynomial of lower degree.


If the two polynomials are, say, f(x) = ax7, and g(x) = -ax7, then the sum is zero, so the product of the degrees would be zero.

Assuming this is not the case and we have, say, f(x) = ax7 + bx + c, and g(x) = -ax7 + dx + e, then the sum is of first degree and the product is 7.


(The above assumes I've interpreted the question correctly!).

 May 23, 2020

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