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Suppose that $f$ is a quadratic polynomial and $g$ is a cubic polynomial, and both $f$ and $g$ have a leading coefficient of $1$. What is the maximum degree of the polynomial $(f(x))^3 - (g(x))^2 + f(x) - 1$?

 

Anyone have an Idea on how to approach this problem?

Guest Jun 16, 2017
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 f(x)  =  x2 + bx + c

g(x)  =  x3 + wx2 + yx + z

 

[f(x)]3 - [g(x)]2 + f(x) - 1

(x2+bx+c)3 - (x3+wx2+yx+z)2   +   x2+bx+c   -   1

(x2+bx+c)(x2​+bx+c)(x2​+bx+c)  -  (x3+wx2+yx+z)(x3+wx2+yx+z)   +   x2+bx+c   -   1

x4 + ...

 

We could continue, but by now I can see that the first term of   [f(x)]3   will be   1x6   and the first term of   [g(x)]2   will be   1x6   . Since   [g(x)]2    is being subtracted,  the   1x6   will cancel each other and go away. The next biggest exponent comes from  x3 * wx2  =  wx5 .

 

So  5   is the maximum degree. 

hectictar  Jun 16, 2017

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