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)

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 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 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.