/*! 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 67 Determine whether each statement... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 67

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Every time I divide polynomials using synthetic division, I am using a highly condensed form of the long division procedure where omitting the variables and exponents does not involve the loss of any essential data.

Short Answer

Expert verified
The statement makes sense because synthetic division is a short-form of polynomial long division, where, although variables and exponents are not explicitly written, their essential contribution to the overall data is not lost.

Step by step solution

01

Assess the validity of the first part of the statement

Let's examine the first claim: 'Every time I divide polynomials using synthetic division, I am using a highly condensed form of the long division procedure.' Synthetic division is indeed a shorthand method of dividing polynomials, offering a more straightforward approach than the long division process. It specifically deals with the division of a polynomial by a binomial of the form \(x - a\). So, the first part of the statement is correct.
02

Assess the validity of the second part of the statement

Now let's consider the latter part of the statement: 'omitting the variables and exponents does not involve the loss of any essential data.' In synthetic division, the variables are indeed omitted, and we deal mainly with the coefficients. The divisors used in synthetic division are written in terms of coefficients rather than variables. And even though we don't explicitly write the variables and exponents, each position in the division process corresponds to a power of \(x\), implicitly keeping track of the exponents. Therefore, although variables and exponents are not visually present, their significance is indirectly retained, which means the statement also holds in the second part.

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

Sound intensity varies inversely as the square of the distance from the sound source. If you are in a movie theater and you change your seat to one that is twice as far from the speakers, how does the new sound intensity compare to that of your original seat?

What is a polynomial inequality?

Explain what is meant by combined variation. Give an example with your explanation.

In this exercise, we lead you through the steps involved in the proof of the Rational Zero Theorem. Consider the polynomial equation $$a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0}=0$$ and let \(\frac{P}{q}\) be a rational root reduced to lowest terms. a. Substitute \(\frac{p}{q}\) for \(x\) in the equation and show that the equation can be written as $$a_{n} p^{n}+a_{n-1} p^{n-1} q+a_{n-2} p^{n-2} q^{2}+\cdots+a_{1} p q^{n-1}=-a_{0} q^{n}$$ b. Why is \(p\) a factor of the left side of the equation? c. Because \(p\) divides the left side, it must also divide the right side. However, because \(\frac{P}{q}\) is reduced to lowest terms, \(p\) and \(q\) have no common factors other than \(-1\) and 1 Because \(p\) does divide the right side and has no factors in common with \(q^{n},\) what can you conclude? d. Rewrite the equation from part (a) with all terms containing \(q\) on the left and the term that does not have a factor of \(q\) on the right. Use an argument that parallels parts (b) and (c) to conclude that \(q\) is a factor of \(a_{n}\).

Whe lise a graphing utility to graph $$ f(x)=\frac{x^{2}-4 x+3}{x-2} \text { and } g(x)=\frac{x^{2}-5 x+6}{x-2} $$ What differences do you observe between the graph of \(f\) and the graph of \(g\) ? How do you account for these differences?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.