/*! 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 29 Perform each division. $$ \f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 29

Perform each division. $$ \frac{-27 r^{4}+36 r^{3}-6 r^{2}-26 r+2}{-3 r} $$

Short Answer

Expert verified
9r^{3} - 12r^{2} + 2r + \frac{26}{3} - \frac{2}{3r}

Step by step solution

01

Set up the division

Write down the polynomial \( -27 r^{4} + 36 r^{3} - 6 r^{2} - 26 r + 2 \) and divide each term by \( -3 r \).
02

Divide each term individually

Divide each term of the polynomial by \( -3 r \): 1. \ \frac{-27 r^{4}}{-3 r} = 9 r^{3} \ 2. \ \frac{36 r^{3}}{-3 r} = -12 r^{2} \ 3. \ \frac{-6 r^{2}}{-3 r} = 2 r \ 4. \ \frac{-26 r}{-3 r} = \frac{26}{3} \ 5. \ \frac{2}{-3 r} = -\frac{2}{3 r} \
03

Combine the results

Combine the results from each term's division: \( 9 r^{3} - 12 r^{2} + 2 r + \frac{26}{3} - \frac{2}{3 r} \).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Division
Polynomial division is a method used to divide a polynomial by another polynomial or a monomial. It is similar to long division but involves algebraic expressions. When dividing polynomials, we break down the terms and handle each term separately. This division method helps simplify complex algebraic expressions and makes solving equations more manageable.
For example, when dividing the polynomial \(-27 r^4 + 36 r^3 - 6 r^2 - 26 r + 2\) by \(-3 r\), we individually divide each term of the polynomial by the divisor \(-3 r\).
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. They can represent numbers and relationships between them. Polynomials are a type of algebraic expression with one or more terms.
For example, the polynomial \(-27 r^4 + 36 r^3 - 6 r^2 - 26 r + 2\) is an algebraic expression with five terms.Each term in a polynomial can have:
  • A coefficient: The numerical part (e.g., -27 in \(-27 r^4\)).
  • A variable: Letters that represent numbers (e.g., \r\ in \(-27 r^4\)).
  • An exponent: The power to which the variable is raised (e.g., 4 in \(-27 r^4\)).
Step-by-Step Solution
To solve polynomial division, follow a systematic, step-by-step approach:
1. Set up the division: Write down the polynomial and divide it by the monomial. In our example, we need to divide each term of \(-27 r^{4} + 36 r^{3} - 6 r^{2} - 26 r + 2\) by \(-3 r\).
2. Divide each term individually. Break it down as follows:
  • \frac{-27 r^{4}}{-3 r} = 9 r^{3}\.
  • \frac{36 r^{3}}{-3 r} = -12 r^{2}\.
  • \frac{-6 r^{2}}{-3 r} = 2 r\.
  • \frac{-26 r}{-3 r} = \frac{26}{3}\.
  • \frac{2}{-3 r} = -\frac{2}{3 r}\.
3. Combine the results: Simply put together the quotients obtained from each term. The result is \9 r^{3} - 12 r^{2} + 2 r + \frac{26}{3} - \frac{2}{3 r}\.
Division of Polynomials by Monomials
Dividing polynomials by monomials helps simplify expressions and solve equations. A monomial is a single-term algebraic expression, like \(-3 r\) in our example. Here are some tips for dividing polynomials by monomials:
  • Divide each term separately: This makes calculations easier and prevents confusion.
  • Simplify fractions: If division results in fractions, simplify them to their lowest terms.
  • Keep track of coefficients, variables, and exponents: Ensure each part of the term is divided correctly.
By following these guidelines, you can smoothly solve problems involving polynomial division, like the given example: \frac{-27 r^{4} + 36 r^{3} - 6 r^{2} - 26 r + 2}{-3 r} = 9 r^{3} - 12 r^{2} + 2 r + \frac{26}{3} - \frac{2}{3 r}\.

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

Use scientific notation to calculate the answer to each problem. Astronomers using the Spitzer Space Telescope discovered a twisted double- helix nebula, a conglomeration of dust and gas stretching across the center of the Milky Way galaxy. This nebula is \(25,000\) light-years from Earth. If one light-year is about \(6,000,000,000,000\) (that is, 6 trillion) miles, about how many miles is the twisted double-helix nebula from Earth?

Perform each indicated operation. Subtract the sum of \(9 t^{3}-3 t+8\) and \(t^{2}-8 t+4\) from the sum of \(12 t+8\) and \(t^{2}-10 t+3\)

Use scientific notation to calculate the answer to each problem. Venus is \(6.68 \times 10^{7} \mathrm{mi}\) from the sun. If light travels at a speed of \(1.86 \times 10^{5} \mathrm{mi}\) per sec, how long does it take light to travel from the sun to Venus?

Perform each division using the "long division" process. $$ \frac{6 p^{4}-16 p^{3}+15 p^{2}-5 p+10}{3 p+1} $$

Perform each division using the "long division" process. $$ \frac{2 r^{3}-5 r^{2}-6 r+15}{r-3} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.