/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 84 Is every rational function a pol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 84

Is every rational function a polynomial function? Is every polynomial function a rational function? Explain.

Short Answer

Expert verified
Yes, every polynomial function is a rational function because a polynomial is a rational function with a denominator of 1. But no, not every rational function is a polynomial function as the denominator in a rational function is not always a constant, as required in a polynomial.

Step by step solution

01

Understand a Rational Function

A rational function is a function that can be written as the ratio of two polynomial functions. To put it more simply, it's a fraction wherein both the numerator and the denominator are polynomials. The general form of a rational function is \( R(x) = \frac{P(x)}{Q(x)} \) where P(x) and Q(x) are polynomial functions.
02

Understand a Polynomial Function

A polynomial function is one made up of terms, which are variables, coefficients, and exponents, where the exponents must be non-negative integers. The general form of a polynomial function is \( P(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + ... + a_nx^n \) where \(a_0, a_1, ..., a_n\) are constants, and n is a non-negative integer.
03

Compare Rational and Polynomial functions

It can be seen that every polynomial function is a rational function. This is because a polynomial can be considered as a rational function with a denominator of 1. But not every rational function is a polynomial function due to the denominator not being a constant. In a rational function, variables could also exist in the denominator, which violates the principles of a polynomial.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the key numbers of the expression. \(3 x^{2}-x-2\)

Determine whether the statement is true or false. Justify your answer. A polynomial can have infinitely many vertical asymptotes.

Find the domain of \(x\) in the expression. Use a graphing utility to verify your result. \(\sqrt{81-4 x^{2}}\)

Write a rational function \(f\) that has the specified characteristics. (There are many correct answers.) (a) Vertical asymptote: \(x=2\) Horizontal asymptote: \(y=0\) Zero: \(x=1\) (b) Vertical asymptote: \(x=-1\) Horizontal asymptote: \(y=0\) Zero: \(x=2\) (c) Vertical asymptotes: \(x=-2, x=1\) Horizontal asymptote: \(y=2\) Zeros: \(x=3, x=-3\), (d) Vertical asymptotes: \(x=-1, x=2\) Horizontal asymptote: \(y=-2\) Zeros: \(x=-2, x=3\)

Find the domain of \(x\) in the expression. Use a graphing utility to verify your result. \(\sqrt{\frac{x}{x^{2}-2 x-35}}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.