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Chapter 1: Problem 22

Perform the indicated divisions. $$\frac{2 x^{n+2}+4 a x^{n}}{2 x^{n}}$$

Short Answer

Expert verified
After simplification, the expression becomes \(x^2 + 2a\).

Step by step solution

01

Identify common factor

Start by identifying any common factors in the numerator that can be simplified with the denominator. In this case, both terms in the numerator include factors of 2 and \(x^n\).
02

Rewrite the numerator

Rewrite the numerator by factoring out the common factor. The expression becomes: \[ \frac{2x^n(x^2) + 2x^n(2a)}{2x^n} \]
03

Simplify by canceling common factors

Cancel the common factor \(2x^n\) from both the numerator and the denominator. This leaves us with: \(x^2 + 2a\) after cancellation.

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Key Concepts

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

Polynomial Division
Polynomial division is a technique that helps simplify algebraic expressions, much like long division. It involves dividing a polynomial (your numerator) by another polynomial (your denominator). This technique is essential in algebra, especially when simplifying complex expressions or solving polynomial equations.
  • To start with polynomial division, identify any common factors in the numerator and denominator. This helps in simplifying the division process early on.
  • The goal is to rewrite the expression such that the division can be easily executed and simplified. In some cases, this may involve dividing each term in the numerator by the term in the denominator.
For example, consider dividing the expression \( \frac{2x^{n+2}+4ax^n}{2x^n} \). First, observe that each term in the numerator has a common factor \(2x^n\). Dividing both terms of the numerator and the whole denominator by this common factor simplifies the expression greatly. This is the essence of polynomial division, making complex expressions easier to handle.
Factoring Polynomials
Factoring is the process of breaking down a polynomial into simpler parts, or factors, that multiply together to give the original polynomial. This is often used as a first step in polynomial division and is a crucial skill in algebra because it simplifies expressions and equations.
  • When you factor a polynomial, you're looking for common terms that appear in each part of the polynomial. Often, this involves pulling out the greatest common factor (GCF) that can simplify the expression.
  • In the exercise, we factored out the common terms \(2x^n\) from the numerator \(2x^{n+2} + 4ax^n\), which allowed us to see the simpler terms more clearly.
By factoring, we simplify the mechanics of the division. Rather than wrestling with larger terms, the expression becomes more manageable and straightforward to work with, paving the way for the next step鈥攕implification.
Simplifying Expressions
Simplifying algebraic expressions involves reducing them to their simplest form. This often means canceling out common factors in both the numerator and denominator, or reducing terms to the lowest possible expression.
  • The purpose of simplification is to make the expression easier to understand and work with. It is essential in solving equations efficiently and correctly.
  • After factoring out \(2x^n\) from each term of the numerator, the expression \( \frac{2x^n(x^2) + 2x^n(2a)}{2x^n} \) simplifies by canceling the common factor \(2x^n\), leading to \(x^2 + 2a\).
This final simplified form is not only easier to interpret but also makes solving and analyzing problems more straightforward. Simplification is an important skill that enables you to work with more complex algebraic functions with greater ease.

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Most popular questions from this chapter

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For computer memory, the metric prefixes have an unusual meaning: \(1 \mathrm{KB}=2^{10}\) bytes, \(1 \mathrm{MB}=2^{10} \mathrm{KB}, 1 \mathrm{GB}=2^{10} \mathrm{MB},\) and I \(\mathrm{TB}=2^{10}\) GB. How many bytes are there in 1 TB? (KB is kilobyte, MB is megabyte, GB is gigabyte, TB is terabyte)

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