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

Verified step by step guidance

1. Start by writing down the given polynomial function: \(f(x) = -x^3 + x^2 + 2x\).
2. Factor the polynomial by first factoring out the greatest common factor (GCF). Identify the GCF of all terms, which is \(-x\), and factor it out: \(f(x) = -x(x^2 - x - 2)\).
3. Next, factor the quadratic expression inside the parentheses: \(x^2 - x - 2\). Find two numbers that multiply to \(-2\) and add to \(-1\). These numbers are \(-2\) and \(1\), so factor as \((x - 2)(x + 1)\).
4. Rewrite the fully factored form of the polynomial as \(f(x) = -x(x - 2)(x + 1)\).
5. Use the factored form to find the roots of the function by setting each factor equal to zero: \(-x = 0\), \(x - 2 = 0\), and \(x + 1 = 0\). These roots will help you plot the x-intercepts on the graph.

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables raised to whole-number exponents and coefficients combined using addition, subtraction, and multiplication. 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 helps identify the roots or zeros of the function, which correspond to the x-intercepts on the graph, making it easier to sketch the function accurately.

Graphing Polynomial Functions

Graphing involves plotting key points such as zeros, intercepts, and turning points, and understanding the end behavior based on the degree and leading coefficient. Factoring aids in finding zeros, while evaluating the function at various points helps shape the curve.