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

 Jul 19, 2019
 #2
avatar+507 
+2

Because both the leading coefficients of f(x) and g(x) are 1, the x^6 term would cancel out, giving a maximum degree of 5.

 Jul 19, 2019
 #3
avatar+104756 
+3

Let g(x)  =  x^3 + x^2  

Let f(x) be  x^2 + x

 

Note that....in both cases.....the lead coefficient is "1"

 

So....we have that

 

[ x^2 + x]^3  -  [ x^3 + x^2 ]^2 + [x^2 + x ] - 1     =  

 

x^5 + 2 x^4 + x^3 + x^2 + x - 1

 

So.....as Davis found....the max degree is  5

 

 

cool cool cool

 Jul 19, 2019

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