Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=2x³-5x²-x+6

Verified step by step guidance

1. Identify the given polynomial function: \(f(x) = 2x^3 - 5x^2 - x + 6\).
2. Attempt to factor the polynomial by grouping. Group the terms as \((2x^3 - 5x^2)\) and \((-x + 6)\).
3. Factor out the greatest common factor (GCF) from each group: from the first group factor out \(x^2\), and from the second group factor out \(-1\), giving \(x^2(2x - 5) - 1(x - 6)\).
4. Check if the binomials inside the parentheses are the same. If not, try rearranging or use another factoring method such as the Rational Root Theorem to find possible roots and factor accordingly.
5. Once factored completely, use the factored form to find the roots (zeros) of the function by setting each factor equal to zero, then plot these roots on the x-axis and analyze the end behavior to sketch 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 a polynomial requires plotting its zeros, determining the end behavior based on degree and leading coefficient, and analyzing the function’s behavior between roots. Factoring first simplifies finding zeros, which are critical points for sketching the curve.