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

Verified step by step guidance

1. Identify the degree of the polynomial by adding the exponents of each factor: $2x^2(x - 1)^3(x + 2)$ has a degree of $2 + 3 + 1 = 6$.
2. Determine the end behavior of the polynomial. Since the leading term is $2x^6$, which is positive and even, the graph will rise to infinity on both ends.
3. Find the x-intercepts by setting each factor equal to zero: $2x^2 = 0$, $(x - 1)^3 = 0$, and $(x + 2) = 0$. This gives x-intercepts at $x = 0$, $x = 1$, and $x = -2$.
4. Analyze the multiplicity of each root: $x = 0$ has multiplicity 2, $x = 1$ has multiplicity 3, and $x = -2$ has multiplicity 1. This means the graph will touch the x-axis at $x = 0$, cross at $x = -2$, and have a point of inflection at $x = 1$.
5. Sketch the graph using the information from the previous steps, ensuring the correct end behavior and behavior at each intercept.

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 graphing them, as it helps identify their degree, leading coefficient, and behavior at infinity.

Factoring and Roots

Factoring a polynomial involves expressing it as a product of its linear factors, which correspond to its roots or x-intercepts. For the function f(x) = 2x^2(x - 1)^3(x + 2), the roots are x = 0 (with multiplicity 2), x = 1 (with multiplicity 3), and x = -2 (with multiplicity 1). Knowing the roots is crucial for graphing, as they indicate where the graph intersects the x-axis and how it behaves near those points.

Graphing Techniques

Graphing polynomial functions requires understanding their key features, including intercepts, end behavior, and turning points. The degree of the polynomial determines the number of turning points, while the leading coefficient influences the direction of the graph as x approaches positive or negative infinity. Techniques such as plotting key points, analyzing symmetry, and using the roots help create an accurate representation of the polynomial's behavior.