Adding which terms to $3 x^2 y$ would result in a monomial? Check all that apply.
$3 x y$
$-12 x^2 y$
$2 x^2 y^2$
$7 x y^2$
$-10 x^2$
$4 x^2 y$
$3 x^3$
Answer (2)
The terms that can be added to 3 x 2 y to result in a monomial are − 12 x 2 y and 4 x 2 y . These terms are identical in their variable factors and exponents to the original term. Thus, the final answer is − 12 x 2 y , 4 x 2 y .
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Identify that only like terms can be combined to form a monomial.
Check each term to see if it has the same variable factors with the same exponents as 3 x 2 y .
Combine like terms: 3 x 2 y + ( − 12 x 2 y ) = − 9 x 2 y and 3 x 2 y + 4 x 2 y = 7 x 2 y .
Conclude that the terms − 12 x 2 y and 4 x 2 y result in a monomial when added to 3 x 2 y , so the answer is − 12 x 2 y , 4 x 2 y .
Explanation
Understanding the Problem We are given the expression 3 x 2 y and asked to identify which terms, when added to it, result in a monomial. A monomial is a single term expression. For the sum to be a monomial, the added term must have the same variable factors with the same exponents as 3 x 2 y . In other words, we are looking for like terms.
Checking Each Term Let's examine each option:
3 x y : This term has x to the power of 1 and y to the power of 1. This is different from x 2 y , so it's not a like term.
− 12 x 2 y : This term has x to the power of 2 and y to the power of 1. This is the same as x 2 y , so it's a like term. Adding it to 3 x 2 y gives 3 x 2 y − 12 x 2 y = − 9 x 2 y , which is a monomial.
2 x 2 y 2 : This term has x to the power of 2 and y to the power of 2. This is different from x 2 y , so it's not a like term.
7 x y 2 : This term has x to the power of 1 and y to the power of 2. This is different from x 2 y , so it's not a like term.
− 10 x 2 : This term has x to the power of 2, but no y . This is different from x 2 y , so it's not a like term.
4 x 2 y : This term has x to the power of 2 and y to the power of 1. This is the same as x 2 y , so it's a like term. Adding it to 3 x 2 y gives 3 x 2 y + 4 x 2 y = 7 x 2 y , which is a monomial.
3 x 3 : This term has x to the power of 3, but no y . This is different from x 2 y , so it's not a like term.
Final Answer Therefore, the terms that, when added to 3 x 2 y , result in a monomial are − 12 x 2 y and 4 x 2 y .
Examples
Understanding monomials and combining like terms is crucial in various real-world applications. For instance, when calculating the area of a rectangular garden with variable side lengths, you might end up with an expression like 3 x 2 y + 4 x 2 y , where x and y represent the dimensions. Combining these like terms simplifies the expression to 7 x 2 y , making it easier to calculate the total area for different values of x and y . This principle extends to more complex scenarios in engineering, physics, and economics, where simplifying expressions helps in modeling and solving problems efficiently.
