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

1 Answer
Alan P.
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(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.

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