/*! 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 16 For Exercises 7 to \(30,\) divid... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 16

For Exercises 7 to \(30,\) divide. $$\frac{a^{3}-5 a^{2}+7 a}{a}$$

Short Answer

Expert verified
The result of dividing the polynomial \(a^3 - 5a^2 + 7a\) by the monomial 'a' is \(a^2 - 5a + 7\).

Step by step solution

01

Identify Terms in the Polynomial and the Monomial to Divide With

Our polynomial has three terms: \(a^3\), \(-5a^2\), and \(7a\). The monomial we are dividing with is 'a'.
02

Divide Each Term of the Polynomial by the Monomial

To divide each term of the polynomial by the monomial, apply the formula for division in each term:\[\frac{a^3}{a} = a^2\]\[\frac{-5a^2}{a} = -5a\]\[\frac{7a}{a} = 7\]These are our new terms after performing the division.
03

Formulate the Final Result

Combine all these new terms to form our final result, which is the polynomial after division by the monomial 'a' : \[a^2 - 5a + 7\]

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.

Dividing Polynomials by Monomials
Polynomial division can often be simplified when we are dividing by a monomial. A monomial is an algebraic expression with only one term, such as 'a' in our exercise. When you divide a polynomial by a monomial, you apply division term-by-term.

Here's a practical approach to dividing polynomials by monomials:
  • Write down the polynomial and the monomial, clearly separating each term of the polynomial.
  • Divide each term of the polynomial by the monomial independently.
  • Use the laws of exponents to simplify each obtained term.
  • Combine the simplified terms to get the final result.
Using these steps, you'll find that each term in the polynomial gets reduced by a power of the variable involved in the monomial, provided the variable is the same. For example, \(a^3\) divided by \(a\) becomes \(a^2\), because you subtract the exponent of the divisor from the exponent of the dividend, based on the laws of exponents.
Algebraic Long Division
Algebraic long division is a method used when dividing polynomials by other polynomials with more than one term. It is similar to the long division method we all learn in elementary arithmetic, but with a twist to accommodate algebraic terms.

Here are the typical steps for algebraic long division:
  • Write the dividend and divisor in the long division format.
  • Divide the leading term of the dividend by the leading term of the divisor.
  • Multiply the entire divisor by the result of the first division and subtract it from the dividend.
  • Bring down the next term of the original dividend and repeat the process until there are no terms left to bring down.
  • If the last remainder is a polynomial of lower degree than the divisor, this is your remainder.
One key point to remember is to carefully align like terms vertically during subtraction to avoid common errors. The quotient from the division process is the simplified form of the original polynomial, while the remaining smaller-degree polynomial is the remainder.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is a fundamental skill in algebra. In doing so, you make the expressions easier to work with by combining like terms and reducing fractions when possible.

To simplify an algebraic expression:
  • Combine like terms, which are terms that have the same variables raised to the same power.
  • Apply the distributive property to remove parentheses if necessary.
  • Reduce any fractions by dividing both the numerator and the denominator by their greatest common factor.
  • Perform any operations that are still possible, such as adding coefficients or constants.
Returning to our exercise, once we divide each term of the polynomial by \(a\) and arrive at \(a^2 - 5a + 7\), no further simplification is required. There are no like terms to combine, and the expression is already in its simplest form.

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

For Exercises, write in scientific notation. $$547,000,000$$

Simplify. $$(x+1)\left(x^{3}-x^{2}+x-1\right)$$

Multiply. $$(2 a+5 b)(7 a-2 b)$$

For Exercises, write in decimal notation. $$1.02 \times 10^{-9}$$

State whether the coefficient of the \(x\) term of the product is positive, negative, or zero. $$(a x+b)^{2}, \text { where } a<0 \text { and } b<0$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

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.