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

Verified step by step guidance

1. Start by examining the polynomial function: \(f(x) = x^3 + 5x^2 - x - 5\). Since it is not factored, the first step is to factor it to make graphing easier.
2. Group the terms to factor by grouping: group the first two terms and the last two terms separately, so you have \((x^3 + 5x^2) + (-x - 5)\).
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 \(x^2(x + 5) - 1(x + 5)\).
4. Notice that \((x + 5)\) is a common binomial factor. Factor it out to get \((x + 5)(x^2 - 1)\).
5. Recognize that \(x^2 - 1\) is a difference of squares, which factors further into \((x - 1)(x + 1)\). So the fully factored form is \((x + 5)(x - 1)(x + 1)\). This factored form helps identify the roots and sketch the graph.

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with 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 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.