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.

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.

To use the leading coefficient test to determine the end behavior of the polynomial function ( f(x) = -5(x^2 + 1)(x - 2) ), we first need to analyze the degree and the leading coefficient of the polynomial.

The degree of the polynomial is the highest exponent of the variable ( x ), which is 3 in this case. The leading coefficient is the coefficient of the term with the highest degree, which is -5.

Based on the degree and the leading coefficient, we can determine the end behavior of the function as follows:

1. For polynomials with odd degree:

   • If the leading coefficient is positive, the function will rise to the left and rise to the right.
   • If the leading coefficient is negative, the function will fall to the left and fall to the right.

2. For polynomials with even degree:

   • If the leading coefficient is positive, the function will rise to the left and rise to the right.
   • If the leading coefficient is negative, the function will fall to the left and rise to the right.

In the given polynomial ( f(x) = -5(x^2 + 1)(x - 2) ), the degree is odd (3) and the leading coefficient is negative (-5). Therefore, according to the leading coefficient test:

• The function will fall to the left and fall to the right.