Making Monomials: A Math Challenge

Hey math whizzes! Today, we're diving into the super cool world of monomials. You know, those algebraic expressions that are just one term? We're going to tackle a fun problem: given the expression $3x^2y$, what can we add to it to keep it as a single monomial? This is a great way to really understand what makes a monomial tick. We'll go through each option, break it down, and figure out why it works or doesn't work. Get ready to flex those math muscles, guys!

Understanding Monomials: The Building Blocks

Alright, let's kick things off with a solid understanding of what a monomial actually is. Think of it as a single, indivisible piece in the grand puzzle of algebra. A monomial is an expression that consists of a single term. This term can be a number (like 7), a variable (like $x$), or a product of numbers and variables raised to non-negative integer powers (like $5x^3y^2$). The key things to remember are: no division by a variable (so no $1/x$), and no negative exponents (so no $x^{-2}$). Our starting point is $3x^2y$. This is already a monomial, with a coefficient of 3 and variables $x$ raised to the power of 2, and $y$ raised to the power of 1. The goal is to add another term to this, and the result must still be a single monomial. This means that when we add, the terms must be like terms. Like terms are terms that have the exact same variables raised to the exact same powers. For example, $2x^2y$ and $5x^2y$ are like terms, but $2x^2y$ and $2xy^2$ are not, because the powers of $x$ and $y$ are different. If we add like terms, we just add their coefficients: $2x^2y + 5x^2y = (2+5)x^2y = 7x^2y$, which is still a monomial! If we add unlike terms, like $3x^2y + 2x^2y^2$, the result is $3x^2y + 2x^2y^2$. This expression has two distinct terms, making it a binomial, not a monomial. So, our mission is to find terms that, when added to $3x^2y$, result in a single term because they are like terms. Let's dive into the options provided and see which ones fit the bill.

Analyzing the Options: Which Ones Make the Cut?

Now for the fun part, guys! We've got our starting monomial, $3x^2y$, and we need to find terms to add to it that will keep the result as a single monomial. Remember, this means the term we add must be a like term to $3x^2y$. A like term has the exact same variable parts: $x^2y$. Let's break down each option:

A. $3xy$: This term has variables $x$ and $y$, but the power of $x$ is 1, not 2. So, $3xy$ is not a like term to $3x^2y$. Adding them would give us $3x^2y + 3xy$, which is a binomial (two terms).

B. $-12x^2y$: Bingo! This term has the variable $x$ raised to the power of 2 and the variable $y$ raised to the power of 1. These are the exact same variable parts as our original term $3x^2y$. So, $-12x^2y$ is a like term. If we add them, we get $3x^2y + (-12x^2y) = (3 - 12)x^2y = -9x^2y$. This is a single monomial! So, Option B is a winner!

C. $2x^2y^2$: This term has $x^2$, which matches, but it also has $y^2$. Our original term has $y^1$. Since the powers of $y$ don't match, $2x^2y^2$ is not a like term. Adding them would give $3x^2y + 2x^2y^2$, a binomial.

D. $7xy^2$: Similar to option C, this term has $x^1$ (not $x^2$) and $y^2$ (not $y^1$). The variable parts don't match our $3x^2y$ at all. So, $7xy^2$ is not a like term. Adding them results in a binomial.

E. $-10x^2$: This term only has the variable $x$ raised to the power of 2. It's missing the $y$ variable that's in our original $3x^2y$. Therefore, $-10x^2$ is not a like term. Adding them gives $3x^2y - 10x^2$, which is a binomial.

F. $4x^2y$: Take a look at this one! It has $x^2$ and $y^1$. These are exactly the same variable parts as $3x^2y$. This means $4x^2y$ is a like term. If we add them, we get $3x^2y + 4x^2y = (3+4)x^2y = 7x^2y$. This is a single monomial! So, Option F is also a winner!

G. $3x^3$: This term has $x^3$, which is different from $x^2$. It's also missing the $y$ variable. So, $3x^3$ is not a like term to $3x^2y$. Adding them results in a binomial.

The Verdict: Putting It All Together

So, after breaking down each option, we found that the terms that can be added to $3x^2y$ and result in a single monomial are the ones that are like terms. This means they must have the identical variable parts: $x^2y$. Looking back at our analysis:

• Option B: $-12x^2y$ fits the bill because it has $x^2y$. Adding it results in $-9x^2y$, a monomial.
• Option F: $4x^2y$ also fits the bill because it has $x^2y$. Adding it results in $7x^2y$, a monomial.

All the other options (A, C, D, E, and G) had different variable parts, meaning they were not like terms. When you add unlike terms, you end up with more than one term, which turns your expression into a binomial or a trinomial, and so on, but definitely not a monomial! It's all about those identical variable powers, people. Keep practicing this, and you'll be a monomial master in no time!

Why This Matters in Algebra

Understanding what makes an expression a monomial, and how adding like terms keeps it a monomial, is super important in algebra, guys. Think about when you're simplifying equations or expressions. If you have something like $5x + 3y + 2x - y$, you combine the like terms: $(5x + 2x) + (3y - y) = 7x + 2y$. The result is a binomial. But if you have $4a^2b + 7a^2b - 2a^2b$, you combine them to get $(4+7-2)a^2b = 9a^2b$, which remains a single monomial. This simplification is a fundamental skill. Knowing when terms can be combined into a single monomial helps you reduce complexity and makes solving problems much easier. It's the foundation for working with polynomials, which are expressions made up of one or more monomials. So, the next time you see an expression, try to identify the monomials within it and see how they can be combined. It’s like solving a puzzle, and the more you practice, the quicker you get at spotting the patterns. This skill is essential for everything from basic equation solving to more advanced calculus. So, keep those math brains buzzing!