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Let's analyze each term to determine if adding it to [tex]\( 3 x^2 y \)[/tex] results in a monomial. A monomial is a single term consisting of a product of constants and variables with non-negative integer exponents.
We start with the base term [tex]\( 3 x^2 y \)[/tex]. This term can be described as having the following powers of the variables:
- [tex]\( x \)[/tex]: exponent is 2
- [tex]\( y \)[/tex]: exponent is 1
We'll go through each term one by one to see if it has the same exponents for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] as the base term. If it does, adding the coefficients would also result in a single term with the same variables and powers, thus still being a monomial.
1. Term: [tex]\( 3 x y \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 1), [tex]\( y \)[/tex] (degree 1)
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( x \)[/tex] exponent is 1, not 2. Therefore, adding this term does not result in a monomial.
2. Term: [tex]\( -12 x^2 y \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 2), [tex]\( y \)[/tex] (degree 1)
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( x \)[/tex] and [tex]\( y \)[/tex] exponents match the base term. Therefore, adding this term does result in a monomial.
3. Term: [tex]\( 2 x^2 y^2 \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 2), [tex]\( y \)[/tex] (degree 2)
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( y \)[/tex] exponent is 2, not 1. Therefore, adding this term does not result in a monomial.
4. Term: [tex]\( 7 x y^2 \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 1), [tex]\( y \)[/tex] (degree 2)
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( x \)[/tex] exponent is 1, not 2. Therefore, adding this term does not result in a monomial.
5. Term: [tex]\( -10 x^2 \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 2), no [tex]\( y \)[/tex] variable present
- Comparison with [tex]\( 3 x^2 y \)[/tex]: There is no [tex]\( y \)[/tex] variable. Therefore, adding this term does not result in a monomial.
6. Term: [tex]\( 4 x^2 y \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 2), [tex]\( y \)[/tex] (degree 1)
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( x \)[/tex] and [tex]\( y \)[/tex] exponents match the base term. Therefore, adding this term does result in a monomial.
7. Term: [tex]\( 3 x^3 \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 3), no [tex]\( y \)[/tex] variable present
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( x \)[/tex] exponent is 3, not 2, and there is no [tex]\( y \)[/tex] variable. Therefore, adding this term does not result in a monomial.
Thus, the terms that, when added to [tex]\( 3 x^2 y \)[/tex], result in a monomial are:
- [tex]\( -12 x^2 y \)[/tex]
- [tex]\( 4 x^2 y \)[/tex]
These correspond to the terms at index positions:
- [tex]\( -12 x^2 y \)[/tex] is at index 1
- [tex]\( 4 x^2 y \)[/tex] is at index 5
Therefore, the indices of the correct terms are [tex]\([1, 5]\)[/tex].
We start with the base term [tex]\( 3 x^2 y \)[/tex]. This term can be described as having the following powers of the variables:
- [tex]\( x \)[/tex]: exponent is 2
- [tex]\( y \)[/tex]: exponent is 1
We'll go through each term one by one to see if it has the same exponents for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] as the base term. If it does, adding the coefficients would also result in a single term with the same variables and powers, thus still being a monomial.
1. Term: [tex]\( 3 x y \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 1), [tex]\( y \)[/tex] (degree 1)
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( x \)[/tex] exponent is 1, not 2. Therefore, adding this term does not result in a monomial.
2. Term: [tex]\( -12 x^2 y \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 2), [tex]\( y \)[/tex] (degree 1)
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( x \)[/tex] and [tex]\( y \)[/tex] exponents match the base term. Therefore, adding this term does result in a monomial.
3. Term: [tex]\( 2 x^2 y^2 \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 2), [tex]\( y \)[/tex] (degree 2)
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( y \)[/tex] exponent is 2, not 1. Therefore, adding this term does not result in a monomial.
4. Term: [tex]\( 7 x y^2 \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 1), [tex]\( y \)[/tex] (degree 2)
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( x \)[/tex] exponent is 1, not 2. Therefore, adding this term does not result in a monomial.
5. Term: [tex]\( -10 x^2 \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 2), no [tex]\( y \)[/tex] variable present
- Comparison with [tex]\( 3 x^2 y \)[/tex]: There is no [tex]\( y \)[/tex] variable. Therefore, adding this term does not result in a monomial.
6. Term: [tex]\( 4 x^2 y \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 2), [tex]\( y \)[/tex] (degree 1)
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( x \)[/tex] and [tex]\( y \)[/tex] exponents match the base term. Therefore, adding this term does result in a monomial.
7. Term: [tex]\( 3 x^3 \)[/tex]
- Variables and exponents: [tex]\( x \)[/tex] (degree 3), no [tex]\( y \)[/tex] variable present
- Comparison with [tex]\( 3 x^2 y \)[/tex]: The [tex]\( x \)[/tex] exponent is 3, not 2, and there is no [tex]\( y \)[/tex] variable. Therefore, adding this term does not result in a monomial.
Thus, the terms that, when added to [tex]\( 3 x^2 y \)[/tex], result in a monomial are:
- [tex]\( -12 x^2 y \)[/tex]
- [tex]\( 4 x^2 y \)[/tex]
These correspond to the terms at index positions:
- [tex]\( -12 x^2 y \)[/tex] is at index 1
- [tex]\( 4 x^2 y \)[/tex] is at index 5
Therefore, the indices of the correct terms are [tex]\([1, 5]\)[/tex].
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