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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 $f(x) + g(x)$?

 Mar 20, 2020
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If you add two polynomials together, you cannot end with a polynomial of higher degree than the highest degree one you started with.

Therefore, the maximum possible degree of the sum of two seven-degree polynomials is seven.

 

However, you can end with a polynomial that has a lower degree; so, if you start with two seven-degree polynomials, you can end with a sum that has any degree from seven down to zero.

 

For example:  if  f(x)  =  3x7 + 16x2 - 9  while  g(x)  =  -3x- 16x2 + 2,  the sum of the two polynomials is -7, a constant polynomial that has degree zero.

 

So, the product would be zero.

 Mar 20, 2020

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