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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)?

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.