Monomial Magic Adding Terms To 3x²y Explained

Mr. Loba Loba · · 6 min read

Table Of Content

  1. Understanding Monomials
  2. The Initial Expression: 3x²y
  3. Analyzing the Options
  4. Conclusion

Hey guys! Let's dive into a fun math problem where we explore how to add terms to an expression to end up with a monomial. A monomial is simply an algebraic expression with only one term. Remember, a term is a product of numbers and variables. So, if we start with 3x²y, what can we add to it to keep it a single term? Let's break it down!

Understanding Monomials

Before we jump into the specifics, let's solidify our understanding of what a monomial actually is. A monomial is an expression that consists of a single term. This term can be a number, a variable, or a product of numbers and variables. The key thing is that there are no addition or subtraction operations involved. For example, 5, x, 3y, and 7x²y are all monomials. On the other hand, x + y, 2x - 3, and 4x² + 1 are not monomials because they involve addition or subtraction.

The coefficients (the numbers) and the variables are multiplied together, and the exponents on the variables must be non-negative integers. This means you can have terms like x² or y³, but not terms like x⁻¹ or √x (which is x¹/²). Understanding this foundation is crucial because when we add terms to our initial expression, 3x²y, we need to make sure that the resulting expression remains a single term to qualify as a monomial. Adding terms with different variable combinations or exponents will break the single-term structure, thus not resulting in a monomial.

The Initial Expression: 3x²y

Our starting point is the term 3x²y. This is a monomial in itself, with a coefficient of 3 and variables x and y. The variable x is raised to the power of 2, and y is raised to the power of 1 (which is usually not explicitly written). To maintain this as a monomial when we add another term, the new term must have the exact same variable part, meaning it must also be x²y. If we add a term with a different variable part, like xy or x²y², we'll end up with an expression with multiple terms, which is no longer a monomial.

Think of it like combining like terms: you can only add or subtract terms that have the same variables raised to the same powers. For example, you can combine 2x²y and 5x²y to get 7x²y, but you can't combine 2x²y and 5xy because they have different variable parts. This concept is fundamental to determining which terms, when added to 3x²y, will result in a monomial. So, our mission is to identify terms that are “like terms” with 3x²y, ensuring that the resulting expression remains a single, unified term. We're essentially looking for terms that fit perfectly into the x²y mold, allowing us to simply adjust the coefficient without altering the fundamental structure of the monomial.

Analyzing the Options

Now, let's examine each of the given options and see if adding them to 3x²y results in a monomial. Remember, the key is that the term we add must have the exact same variable part (x²y) to combine into a single term.

  1. 3xy: This term has xy as the variable part, which is different from x²y. So, adding 3xy to 3x²y would result in a binomial (two terms), not a monomial. For instance, 3x²y + 3xy cannot be simplified into a single term because x²y and xy are not like terms.
  2. -12x²y: This term has the same variable part, x²y, as our initial term. Therefore, adding -12x²y to 3x²y will result in a monomial. The resulting term would be 3x²y - 12x²y = -9x²y, which is a single term. This option works!
  3. 2x²y²: The variable part here is x²y², which is different from x²y. Adding this term would create a binomial: 3x²y + 2x²y². Since these terms are not like terms, they cannot be combined, and the expression remains two terms.
  4. 7xy²: This term has xy² as the variable part, which is also different from x²y. Adding 7xy² to 3x²y results in another binomial: 3x²y + 7xy². These terms are not like terms, so they cannot be combined.
  5. -10x²: Here, the variable part is x², which is different from x²y. Adding this to our initial term gives us 3x²y - 10x², a binomial since the terms cannot be combined.
  6. 4: This is a constant term, and it definitely doesn't have the x²y variable part. Adding 4 to 3x²y gives us 3x²y + 4, a binomial that cannot be simplified further.
  7. 3x³: The variable part here is x³, which is different from x²y. Adding 3x³ to 3x²y results in a binomial: 3x²y + 3x³. These terms are not like terms and cannot be combined.

Conclusion

So, after analyzing all the options, only one term, -12x²y, when added to 3x²y, results in a monomial. This is because it's the only term with the exact same variable part (x²y), allowing us to combine the terms into a single term. Remember, the key to maintaining a monomial when adding terms is to ensure that the terms are “like terms,” meaning they have the same variables raised to the same powers. Hope this helps you guys understand monomials a little better!

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