In Exercises 25–26, graph each polynomial function. f(x) = -x^3(x + 4)^2(x-1)

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1. Identify the degree of the polynomial by adding the exponents of each factor: \(-x^3(x + 4)^2(x-1)\) has a degree of 3 + 2 + 1 = 6.
2. Determine the end behavior of the polynomial. Since the leading term is \(-x^6\), the graph will fall to the left and fall to the right.
3. Find the x-intercepts by setting each factor equal to zero: \(-x^3 = 0\), \((x + 4)^2 = 0\), and \((x - 1) = 0\). This gives x-intercepts at x = 0, x = -4, and x = 1.
4. Analyze the multiplicity of each root: x = 0 has multiplicity 3 (cubic), x = -4 has multiplicity 2 (quadratic), and x = 1 has multiplicity 1 (linear). This affects how the graph behaves at each intercept.
5. Sketch the graph using the intercepts and end behavior. At x = 0, the graph will touch and turn around due to the odd multiplicity. At x = -4, the graph will bounce off the x-axis due to the even multiplicity. At x = 1, the graph will cross the x-axis.

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 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 analyzing their behavior, including their roots and end behavior.

Factoring and Roots

Factoring a polynomial involves expressing it as a product of its linear factors, which directly relates to finding its roots or x-intercepts. For the function f(x) = -x^3(x + 4)^2(x - 1), the roots are x = 0, x = -4, and x = 1. The multiplicity of each root affects the graph's behavior at those points, such as whether the graph crosses or touches the x-axis.

Graphing Techniques

Graphing polynomial functions requires understanding key features such as intercepts, end behavior, and turning points. The degree of the polynomial indicates the maximum number of turning points, while the leading coefficient determines the direction of the graph's ends. By plotting the roots and analyzing the function's behavior around them, one can create an accurate representation of the polynomial's graph.