Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=2x³(x²-4)(x-1)

Verified step by step guidance

1. Identify the given polynomial function: \(f(x) = 2x^3(x^2 - 4)(x - 1)\).
2. Recognize that the polynomial is already partially factored, but \(x^2 - 4\) is a difference of squares and can be factored further as \(x^2 - 4 = (x - 2)(x + 2)\).
3. Rewrite the function with the fully factored form: \(f(x) = 2x^3 (x - 2)(x + 2)(x - 1)\).
4. Determine the zeros of the function by setting each factor equal to zero: \(x = 0\), \(x = 2\), \(x = -2\), and \(x = 1\). These are the x-intercepts of the graph.
5. Analyze the multiplicity of each zero: \(x=0\) has multiplicity 3 (since \(x^3\)), which affects the shape of the graph at that intercept, while the others have multiplicity 1. Use this information to sketch the graph, noting where the graph crosses or touches the x-axis.

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding the degree and leading coefficient helps predict the general shape and end behavior of the graph.

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of simpler polynomials or factors. This process reveals the roots or zeros of the function, which correspond to the x-intercepts on the graph, making it easier to plot the function accurately.

Graphing Polynomial Functions

Graphing involves plotting key points such as zeros, intercepts, and analyzing end behavior based on the degree and leading coefficient. Factored form helps identify zeros and their multiplicities, which affect how the graph crosses or touches the x-axis.