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