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?
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.