Questions: Use the given function to complete parts (a) through (e) below. f(x)=x^4-1 x^2 a) Use the Leading Coefficient Test to determine the graph's end behavior. A. The graph of f(x) falls left and rises right. B. The graph of f(x) rises left and rises right. C. The graph of f(x) rises left and falls right. D. The graph of f(x) falls left and falls right: b) Find the x-intercepts. x= (Type an integer or a decimal. Use a comma to separate answers as needed.)

Solution Steps

a) To determine the end behavior of the polynomial function \( f(x) = x^4 - x^2 \), we use the Leading Coefficient Test. The highest degree term is \( x^4 \), which has a positive leading coefficient. For even-degree polynomials with a positive leading coefficient, the graph rises to the left and rises to the right.

b) To find the x-intercepts, we set \( f(x) = 0 \) and solve for \( x \). This involves factoring the polynomial and solving the resulting equations.

The function is given by \( f(x) = x^4 - x^2 \). The highest degree term is \( x^4 \), which has a positive leading coefficient of \( 1 \). Since the degree is even and the leading coefficient is positive, the end behavior of the graph is that it rises to the left and rises to the right.

To find the x-intercepts, we set \( f(x) = 0 \): \[ x^4 - x^2 = 0 \] Factoring gives: \[ x^2(x^2 - 1) = 0 \] This can be further factored as: \[ x^2(x - 1)(x + 1) = 0 \] Setting each factor to zero, we find the x-intercepts: \[ x^2 = 0 \quad \Rightarrow \quad x = 0 \] \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \] Thus, the x-intercepts are \( x = -1, 0, 1 \).

Final Answer