Which Terms, When Added To 3x²y, Result In A Monomial? Options: 3xy, -12x²y, 2x²y², 7xy², -10x², 4x²y.

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In mathematics, a monomial is an algebraic expression consisting of only one term. A term is a product of constants and variables raised to non-negative integer exponents. Understanding monomials is fundamental to grasping more complex algebraic concepts such as polynomials and algebraic manipulation. This article delves into the specifics of identifying which terms, when added to the monomial 3x²y, will result in another monomial. We will explore the essential characteristics of monomials and how like terms combine to maintain the single-term structure. This exploration includes detailed explanations and examples, ensuring a comprehensive understanding of the topic. Let’s begin by revisiting the fundamental definition of a monomial and then systematically examining the given options to determine which ones fit the criteria.

Before diving into the specifics of the problem, it's crucial to define what constitutes a monomial. A monomial is an algebraic expression that consists of a single term. This term is a product of constants and variables, where the variables are raised to non-negative integer exponents. For instance, 5x², -3y, and 7xy² are all monomials because they each consist of a single term. In contrast, expressions like 2x + 1 or x² - 3x + 2 are not monomials; they are polynomials, specifically binomials and trinomials, respectively, because they contain more than one term. The absence of addition or subtraction operations between terms is a defining feature of monomials.

Key characteristics of a monomial include:

  1. Single Term: A monomial has only one term.
  2. Constants and Variables: It consists of constants (numbers) and variables (letters representing unknown values).
  3. Non-Negative Integer Exponents: The variables are raised to powers that are non-negative integers (0, 1, 2, 3, ...).

Understanding these characteristics is essential for identifying monomials and for performing algebraic operations involving them. When we add terms to a monomial, we must ensure that the result remains a single term to maintain its monomial status. This typically involves combining like terms, which are terms that have the same variables raised to the same powers. For example, 3x²y and 4x²y are like terms because they both have the variables x raised to the power of 2 and y raised to the power of 1. Adding these terms results in (3 + 4)x²y = 7x²y, which is still a monomial.

Now, let's consider the given monomial 3x²y and examine which of the provided terms, when added to it, will result in a monomial. To achieve this, the added term must be a like term with 3x²y. Like terms have the same variables raised to the same powers, allowing them to be combined into a single term. We will analyze each option individually to determine if it meets this criterion. This process involves comparing the variables and their exponents in each term to those in 3x²y. If the variables and exponents match, the terms are like terms, and their sum will be a monomial. If they do not match, the sum will be a polynomial with multiple terms.

Option A: 3xy

The term 3xy has the variables x and y, but the exponents are different from those in 3x²y. In 3xy, x is raised to the power of 1, and y is raised to the power of 1. In 3x²y, x is raised to the power of 2, and y is raised to the power of 1. Since the exponents of x are not the same, 3xy is not a like term with 3x²y. Therefore, adding 3xy to 3x²y results in the expression 3x²y + 3xy, which has two terms and is not a monomial. The presence of both an x²y term and an xy term means that the expression is a binomial, not a monomial.

Option B: -12x²y

The term -12x²y has the same variables (x and y) raised to the same powers as 3x²y. Both terms have x raised to the power of 2 and y raised to the power of 1. Therefore, -12x²y is a like term with 3x²y. Adding these terms results in:

3x²y + (-12x²y) = (3 - 12)x²y = -9x²y

The result, -9x²y, is a single term and thus a monomial. This is because the coefficients of the like terms are combined while the variables and their exponents remain the same, preserving the single-term structure.

Option C: 2x²y²

The term 2x²y² has the variable x raised to the power of 2, which matches 3x²y. However, y is raised to the power of 2 in 2x²y², whereas it is raised to the power of 1 in 3x²y. Since the exponents of y are different, 2x²y² is not a like term with 3x²y. Adding 2x²y² to 3x²y results in the expression 3x²y + 2x²y², which has two terms and is not a monomial. This expression is a binomial because it contains two distinct terms that cannot be combined further.

Option D: 7xy²

The term 7xy² has the variables x and y, but the exponents differ from those in 3x²y. In 7xy², x is raised to the power of 1, and y is raised to the power of 2. In 3x²y, x is raised to the power of 2, and y is raised to the power of 1. Since the exponents of both x and y are different, 7xy² is not a like term with 3x²y. Adding 7xy² to 3x²y results in the expression 3x²y + 7xy², which has two terms and is not a monomial. This expression represents a binomial, as it consists of two unlike terms.

Option E: -10x²

The term -10x² only has the variable x raised to the power of 2. It does not contain the variable y, which is present in 3x²y. Therefore, -10x² is not a like term with 3x²y. Adding -10x² to 3x²y results in the expression 3x²y - 10x², which has two terms and is not a monomial. The absence of the y variable in -10x² makes it impossible to combine with 3x²y.

Option F: 4x²y

The term 4x²y has the same variables (x and y) raised to the same powers as 3x²y. Both terms have x raised to the power of 2 and y raised to the power of 1. Therefore, 4x²y is a like term with 3x²y. Adding these terms results in:

3x²y + 4x²y = (3 + 4)x²y = 7x²y

The result, 7x²y, is a single term and thus a monomial. The like terms combine to form a single term, maintaining the monomial structure.

In summary, when adding terms to 3x²y, the resulting expression will be a monomial only if the added term is a like term. From the given options, only -12x²y and 4x²y are like terms with 3x²y. Adding these terms results in monomials:

  • 3x²y + (-12x²y) = -9x²y
  • 3x²y + 4x²y = 7x²y

Therefore, the correct answers are B. -12x²y and F. 4x²y. Understanding the concept of like terms is crucial in algebraic operations, ensuring that expressions remain in their simplest form. This exercise highlights the importance of recognizing and combining like terms to maintain the monomial structure. Mastering these basic algebraic principles is essential for tackling more complex mathematical problems in the future.

The terms that, when added to 3x²y, result in a monomial are:

  • B. -12x²y
  • F. 4x²y