Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=-x(x+1)(x-1)

Verified step by step guidance

1. Identify the polynomial function: \( f(x) = -x(x+1)(x-1) \).
2. Recognize that the polynomial is already in factored form: \( -x(x+1)(x-1) \).
3. Determine the roots of the polynomial by setting each factor equal to zero: \( x = 0, x = -1, x = 1 \).
4. Identify the degree of the polynomial by counting the number of factors: The degree is 3, which is odd, indicating the end behavior of the graph.
5. Sketch the graph using the roots and end behavior: The graph will cross the x-axis at \( x = 0, x = -1, \) and \( x = 1 \), and since the leading coefficient is negative, the graph will start in the positive direction and end in the negative direction.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where 'n' is a non-negative integer and 'a_n' are constants. Understanding the structure of polynomial functions is essential for graphing them accurately.

Factoring Polynomials

Factoring a polynomial involves expressing it as a product of its simpler polynomial factors. This process is crucial for identifying the roots or x-intercepts of the polynomial, which are the values of x that make the polynomial equal to zero. For example, the polynomial f(x) = -x(x+1)(x-1) is already factored, making it easier to graph by identifying its roots at x = 0, x = -1, and x = 1.

Graphing Polynomial Functions

Graphing polynomial functions requires plotting points based on the function's values and understanding its behavior at various intervals. Key features to consider include the x-intercepts (roots), y-intercept, and the end behavior of the graph, which is influenced by the leading term of the polynomial. For the function f(x) = -x(x+1)(x-1), the graph will cross the x-axis at its roots and will open downwards due to the negative leading coefficient.