Verified Answer :
- Identify that a sum is a monomial only if the terms added are "like terms" sharing the same variables and exponents.
- Analyze each option to find those with the variable part \(x^2y\).
- Determine that only \(-12x^2y\) and \(4x^2y\) are like terms with \(3x^2y\).
- The correct terms are \(\boxed{-12x^2y, 4x^2y}\).
Explanation
- Understanding Monomials and Like Terms
To determine which terms added to \(3x^2y\) result in a single monomial, we must first recall that a monomial is a single-term algebraic expression. When adding two terms, the sum is a single monomial if and only if the terms are 'like terms'. Like terms are defined as terms containing the exact same variables raised to the exact same exponents. In this case, our base term is \(3x^2y\), which has the variable part \(x^2y\). Therefore, for the addition of any other term \(T\) to \(3x^2y\) to result in a single monomial, \(T\) must be a like term, meaning it must have the variable part \(x^2y\). If \(T\) contains different powers of \(x\) or \(y\), the resulting sum will be a binomial (e.g., \(3x^2y + 7xy^2\)), which is not a single monomial.
- Analyzing the Options
We examine each option to see if it is a like term with \(3x^2y\) (i.e., whether its variable part is \(x^2y\)):
- \(3xy\): The variable part is \(xy\), which is not \(x^2y\).
- \(-12x^2y\): The variable part is \(x^2y\). This is a like term.
- \(2x^2y^2\): The variable part is \(x^2y^2\), which is not \(x^2y\).
- \(7xy^2\): The variable part is \(xy^2\), which is not \(x^2y\).
- \(-10x^2\): The variable part is \(x^2\), which is not \(x^2y\).
- \(4x^2y\): The variable part is \(x^2y\). This is a like term.
- \(3x^3\): The variable part is \(x^3\), which is not \(x^2y\).
- Conclusion and Verification
From the analysis, the terms that have the variable part \(x^2y\) and thus form a monomial when added to \(3x^2y\) are \(-12x^2y\) and \(4x^2y\).
Adding \(-12x^2y\): \(3x^2y + (-12x^2y) = (3 - 12)x^2y = -9x^2y\), which is a monomial.
Adding \(4x^2y\): \(3x^2y + 4x^2y = (3 + 4)x^2y = 7x^2y\), which is a monomial.
The other options result in binomials, such as \(3x^2y + 3xy\). Thus, the correct choices are \(-12x^2y\) and \(4x^2y\).
Examples
Think of monomials like different types of fruit in a basket. If you have 3 apples (represented by \(3x^2y\)), you can only combine them with more apples to keep the "type of fruit" count as a single group. If you add 4 apples (\(4x^2y\)), you get 7 apples, still a single group. But if you try to add 3 oranges (\(3xy\)), you end up with "3 apples and 3 oranges" in your basket—a mix, or a binomial, not a single type anymore. This principle is fundamental in algebra for simplifying expressions by combining like terms, helping you keep your work tidy and organized!