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
