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Suppose that $f(x)$ is a polynomial that has degree $6$ and $g(x)$ is a polynomial that has degree $3$. If $h(x)$ is also a polynomial such that $f(g(x)) + g(h(x)) + h(f(x))$ is a polynomial of degree $36$, then what is the degree of the polynomial $h$?

michaelcai  Sep 16, 2017
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Suppose that $f(x)$ is a polynomial that has degree $6$ and $g(x)$ is a polynomial that has degree $3$. If $h(x)$ is also a polynomial such that $f(g(x)) + g(h(x)) + h(f(x))$ is a polynomial of degree $36$, then what is the degree of the polynomial $h$?

 

If  f is of degree 6 and  g is of degree 3, then the greatest degree of either f(g(x))  or g(f(x)) wil be 18

But addding these together will only result in a polynomial of degree 18  [ at most ]

 

So......   h ( f(x))  must itself be of degree 36 which implies that h will be of degree 6

 

 

cool cool cool

CPhill  Sep 16, 2017

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