Question

How do you use the leading coefficient test to determine the end behavior of the polynomial function `f(x)= -5(x2+1)(x-2)`?

Answer 1

If the leading coefficient is negative, the polynomial function will eventually decrease to negative infinity; if the leading coefficient is positive, the polynomial function will eventually increase to positive infinity.

This answer assumes by leading coefficient you mean the coefficient of the highest powered `x` term (the normal usage).

If `g(x)` is a polynomial with greatest degree `n`
then if `m > n`, the absolute value of `x^m` will be greater than the absolute value of `g(x)` once `x` becomes sufficiently large.

For example if `g(x) = 5x^2 - 4x +12`
`x^3` will have an absolute value greater than `g(x)` provided x is greater than 3.

Since the term with the highest exponent will eventually provide a value that exceeds the combined value of all other terms, the sign of that term determines the eventual direction of the function value.