/*! 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 97 Explain why a polynomial functio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 97

Explain why a polynomial function of degree 20 cannot cross the \(x\) -axis exactly once.

Short Answer

Expert verified
A polynomial function of degree 20 cannot cross the x-axis exactly once because by definition, a polynomial function can crosses the x-axis at most as many times as its degree. Therefore, our polynomial function can cross the x-axis at most 20 times but not exactly once.

Step by step solution

01

Understanding the Basics of a Polynomial Function of a Given Degree

A polynomial function is a mathematical equation involving a variable and coefficients. The highest power of the variable in the equation denotes the degree of the polynomial function. A polynomial function of degree \(n\) can have at most \(n\) real roots meaning it crosses the x-axis at most \(n\) times.
02

Relating Degree and Zeroes of the Given Polynomial Function

We're given a polynomial function of degree 20. Following our understanding from previous step, this function might have at most 20 real roots. Hence, it means it could cross the x-axis at most 20 times.
03

Concluding Why the Polynomial Function Cannot Cross the x-axis Exactly Once

Since a polynomial function of degree 20 has at most 20 real roots, it implies that the function can cross the x-axis a maximum of 20 times. Therefore, it is not possible for the function to cross the x-axis exactly once. It could cross the x-axis fewer times than 20 but not exactly one.

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\).

Find the horizontal asymptote, if there is one, of the graph of rational function. $$h(x)=\frac{12 x^{3}}{3 x^{2}+1}$$

Exercises \(61-63\) will help you prepare for the material covered in the first section of the next chapter. Use point plotting to graph \(f(x)=2^{x}\). Begin by setting up a partial table of coordinates, selecting integers from -3 to \(3,\) inclusive, for \(x .\) Because \(y=0\) is a horizontal asymptote, your graph should approach, but never touch, the negative portion of the \(x\) -axis.

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{(x-3)^{2}}+1$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Decision 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.