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

Feb 9, 2015

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 $g \left(x\right)$ 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 $g \left(x\right)$ once $x$ becomes sufficiently large.

For example if $g \left(x\right) = 5 {x}^{2} - 4 x + 12$
${x}^{3}$ will have an absolute value greater than $g \left(x\right)$ 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.