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

#1**+1 **

f(x) = x^{2} + bx + c

g(x) = x^{3} + wx^{2} + yx + z

[f(x)]^{3} - [g(x)]^{2} + f(x) - 1

(x^{2}+bx+c)^{3} - (x^{3}+wx^{2}+yx+z)^{2} + x^{2}+bx+c - 1

(x^{2}+bx+c)(x^{2}+bx+c)(x^{2}+bx+c) - (x^{3}+wx^{2}+yx+z)(x^{3}+wx^{2}+yx+z) + x^{2}+bx+c - 1

x^{4} + ...

We could continue, but by now I can see that the first term of [f(x)]^{3} will be 1x^{6} and the first term of [g(x)]^{2} will be 1x^{6} . Since [g(x)]^{2 }is being subtracted, the 1x^{6} will cancel each other and go away. The next biggest exponent comes from x^{3} * wx^{2} = wx^{5} .

So 5 is the maximum degree.

hectictar
Jun 16, 2017