Expression Showing Sum Of Polynomials With Like Terms Grouped

Introduction

In the realm of mathematics, particularly in algebra, dealing with polynomials is a fundamental skill. Polynomials are algebraic expressions consisting of variables and coefficients, combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents. When working with polynomials, one often needs to simplify them by combining like terms. Like terms are terms that have the same variables raised to the same powers. This article delves into the process of identifying and grouping like terms within a polynomial expression. We will dissect a given polynomial to demonstrate how to correctly rearrange and group like terms, a crucial step in simplifying and performing operations on polynomials. Mastering this skill is essential for success in higher-level mathematics and related fields.

Understanding Polynomials and Like Terms

To effectively address the question of which expression correctly groups like terms, it's crucial to first define and understand what polynomials and like terms are. A polynomial is an expression consisting of variables (also called unknowns) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For instance, the expression ${10x^2y + 2xy^2 - 4x^2 - 4x^2y}$ is a polynomial. Each part of the polynomial separated by addition or subtraction is called a term. Therefore, in our example, the terms are ${10x^2y}$, ${2xy^2}$, ${-4x^2}$, and ${-4x^2y}$.

Like terms, on the other hand, are terms that have the same variables raised to the same powers. This means that the variables and their respective exponents must match exactly for terms to be considered “like.” For example, ${3x^2}$ and ${-5x^2}$ are like terms because they both have the variable ${x}$ raised to the power of 2. However, ${3x^2}$ and ${3x^3}$ are not like terms because the exponents of ${x}$ are different. Similarly, ${2xy^2}$ and ${2x^2y}$ are not like terms because, although they have the same variables, the exponents are mismatched; in the first term, ${y}$ is squared, while in the second term, ${x}$ is squared.

Identifying like terms is a fundamental step in simplifying polynomials. By grouping like terms together, we can combine their coefficients and reduce the polynomial to its simplest form. This not only makes the polynomial easier to work with but also aids in solving algebraic equations and understanding the behavior of functions represented by polynomials.

Analyzing the Given Polynomial Expression

The polynomial expression we are given is:

${10x^2y + 2xy^2 - 4x^2 - 4x^2y}$

To determine which expression correctly groups the like terms, we first need to identify the like terms within the given polynomial. Let's break down each term and analyze its variables and exponents:

1. ${10x^2y}$: This term has the variable ${x}$ raised to the power of 2 and the variable ${y}$ raised to the power of 1.
2. ${2xy^2}$: This term has the variable ${x}$ raised to the power of 1 and the variable ${y}$ raised to the power of 2. Notice that this term is different from the first term because the exponents of ${x}$ and ${y}$ are swapped.
3. ${-4x^2}$: This term has the variable ${x}$ raised to the power of 2. There is no ${y}$ variable in this term.
4. ${-4x^2y}$: This term has the variable ${x}$ raised to the power of 2 and the variable ${y}$ raised to the power of 1. This term is similar to the first term in that it contains both ${x^2}$ and ${y}$, making them like terms.

Now that we have analyzed each term, we can identify the like terms. The terms ${10x^2y}$ and ${-4x^2y}$ are like terms because they both have ${x^2y}$. The term ${2xy^2}$ is unique and does not have any like terms in the expression. The term ${-4x^2}$ is also unique, as it only contains ${x^2}$ and no ${y}$.

Therefore, when grouping the like terms together, we should group ${10x^2y}$ and ${-4x^2y}$ together, while keeping ${2xy^2}$ and ${-4x^2}$ separate.

Evaluating the Options

Now, let's examine the given options to determine which one correctly groups the like terms together.

Option A: ${(-4x^2) + (-4x^2y) + 10x^2y + 2xy^2}$

In this option, the terms ${-4x^2y}$ and ${10x^2y}$ are grouped together, which is correct. Additionally, the terms ${-4x^2}$ and ${2xy^2}$ are separate, which is also correct. Therefore, Option A correctly groups the like terms together.

Option B: ${10x^2y + 2xy^2 + [(-4x^2) + (-4x^2y)]}$

In this option, ${-4x^2}$ and ${-4x^2y}$ are grouped together. However, these are not like terms. The term ${-4x^2}$ has only ${x}$ squared, while ${-4x^2y}$ has both ${x}$ squared and ${y}$. Therefore, Option B incorrectly groups the terms.

Option C: ${-4x^2 + 2xy^2}$

Option C only presents a portion of the polynomial and does not include all the terms. It does not show the sum of the polynomials with like terms grouped together. Therefore, Option C is incorrect.

Conclusion

After analyzing the given options, we can conclude that Option A is the correct expression that shows the sum of the polynomials with like terms grouped together. This is because it correctly groups the like terms ${10x^2y}$ and ${-4x^2y}$ while keeping the unlike terms ${-4x^2}$ and ${2xy^2}$ separate. Understanding how to identify and group like terms is a fundamental skill in algebra, essential for simplifying polynomials and solving more complex mathematical problems. This exercise highlights the importance of carefully examining the variables and their exponents when working with polynomial expressions.

In summary, mastering the art of grouping like terms in polynomials is a cornerstone of algebraic manipulation. It not only simplifies expressions but also lays the groundwork for more advanced mathematical concepts. By meticulously identifying terms with identical variable compositions and exponents, we can effectively combine them, leading to clearer and more manageable polynomial expressions. This skill is not just an academic exercise; it is a practical tool that enhances problem-solving abilities in various fields, including engineering, computer science, and economics.