Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=-x^3+x^2+2x

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 of a polynomial in one variable 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 degree and leading coefficient of the polynomial is crucial for graphing and analyzing its behavior.

Factoring Polynomials

Factoring a polynomial involves expressing it as a product of its simpler polynomial factors. This process is essential for finding the roots of the polynomial, which are the x-values where the function equals zero. Techniques for factoring include grouping, using the quadratic formula, or applying special product formulas like the difference of squares. Factoring simplifies the graphing process by identifying key points on the x-axis.

Graphing Techniques

Graphing a polynomial function requires understanding its key features, such as intercepts, turning points, and end behavior. After factoring, one can find the x-intercepts (roots) and y-intercept, which help in sketching the graph. Additionally, analyzing the degree of the polynomial informs the number of turning points and the direction in which the graph extends as x approaches positive or negative infinity. These techniques are vital for accurately representing the function visually.