# Rational functions, polynomials

Explain what makes a function a polynomial. Give an example of a function that is a polynomial and a function that is not a polynomial.

Given the polynomial f(x) = 2x3 + 5x2 - x - 5, answer the following questions:

Graph the polynomial. What is the degree of the polynomial?

Explain the constant in the polynomial and how it relates to the graph.

Define the x-intercepts of the polynomial.

Explain the relative maximum and relative minimum of the polynomial in the interval [-3,2]. Is there a different between the relative maximum and minimum and the absolute maximum and minimum?

Part II:

What is a rational function? Give an example. For your example, answer or perform the following:

Graph the rational function.

Give the domain and range of the rational function.

Find the vertical and horizontal asymptotes of the rational function.

https://brainmass.com/math/graphs-and-functions/rational-functions-polynomials-253851

#### Solution Preview

A function f(x) is a polynomial if it has the form

a_0 + a_1 x + a_2 x^2 + ... + a_n x^n

where a_0, a_1, a_2, .., a_n are numbers.

Given the polynomial f(x) = 2x^3 + 5x^2 - x - 5

(a) visit

http://www.walterzorn.com/grapher/grapher_e.htm

and enter 2x^3 + 5x^2 - x - 5

in the box

(b) the polynomial has degree 3, because 3 is the largest exponent on x

(c) the number at the ...

#### Solution Summary

The provides examples of relating functions and polynomials and examples of working with a rational function.

To understand Polynomials and rational functions - To explore the versatility of rational functions, choose a second-order/third-order (e.g., x2/x3) and a third-order/second-order (e.g., x3/x2) rational function. ...

To explore the versatility of rational functions, choose a second-order/third-order (e.g., x2/x3) and a third-order/second-order (e.g., x3/x2) rational function.

To explore the versatility of rational functions, choose a second-order/third-order (e.g., x2/x3) and a third-order/second-order (e.g., x3/x2) rational function. Provide a graph for the second-order rational function (e.g., x2), choosing x values in the range from -10 through +10.

Then, provide at least three variations of the function plotted on the same graph. Include separate changes to a coefficient in the numerator, to a coefficient in the denominator, and to a constant. The changes should be increases or decreases of a factor of 2 in each case. Repeat the procedure making a second graph for the third-order rational function (e.g., x3). For each of the two graphs, describe how changes in coefficients and constants change the behavior of the function.

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