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Answer :
To solve the problem of finding like terms in the polynomial [tex]\(-x^2 + 3x + 4x^3 - 2x^3 + 2x + 7x^2\)[/tex], we need to group terms with the same variables and exponents.
First, let's identify all the terms in the polynomial:
- [tex]\(-x^2\)[/tex]
- [tex]\(3x\)[/tex]
- [tex]\(4x^3\)[/tex]
- [tex]\(-2x^3\)[/tex]
- [tex]\(2x\)[/tex]
- [tex]\(7x^2\)[/tex]
Grouping the like terms:
1. Group terms with [tex]\(x^2\)[/tex]:
- [tex]\(-x^2\)[/tex]
- [tex]\(7x^2\)[/tex]
2. Group terms with [tex]\(x\)[/tex]:
- [tex]\(3x\)[/tex]
- [tex]\(2x\)[/tex]
3. Group terms with [tex]\(x^3\)[/tex]:
- [tex]\(4x^3\)[/tex]
- [tex]\(-2x^3\)[/tex]
Next, we will match these groups with the provided options:
- Option A: [tex]\(-x^2\)[/tex] and [tex]\(4x^3\)[/tex]: These are not like terms, because one is [tex]\(x^2\)[/tex] and the other is [tex]\(x^3\)[/tex].
- Option B: [tex]\(3x\)[/tex], [tex]\(7x^2\)[/tex], and [tex]\(-2x^3\)[/tex]: These are not all like terms; each term has a different variable exponent.
- Option C: [tex]\(2x\)[/tex], [tex]\(-x^2\)[/tex], and [tex]\(4x^3\)[/tex]: These are not like terms for the same reason as Option B.
- Option D: [tex]\(7x^2\)[/tex] and [tex]\(-2x^3\)[/tex]: These are not like terms because they have different exponents.
- Option E: [tex]\(-x^2\)[/tex] and [tex]\(7x^2\)[/tex]: These are like terms because they both have [tex]\(x^2\)[/tex].
- Option F: [tex]\(2x\)[/tex] and [tex]\(3x\)[/tex]: These are like terms because they both have [tex]\(x\)[/tex].
- Option G: [tex]\(2x\)[/tex], [tex]\(3x\)[/tex], [tex]\(-x^2\)[/tex], and [tex]\(7x^2\)[/tex]: This option contains two sets of like terms: [tex]\(2x\)[/tex] and [tex]\(3x\)[/tex]; [tex]\(-x^2\)[/tex] and [tex]\(7x^2\)[/tex].
- Option H: [tex]\(4x^3\)[/tex] and [tex]\(-2x^3\)[/tex]: These are like terms because they both have [tex]\(x^3\)[/tex].
After this careful grouping and analysis, we can see that the correct answers are:
- E: [tex]\(-x^2\)[/tex] and [tex]\(7x^2\)[/tex]
- F: [tex]\(2x\)[/tex] and [tex]\(3x\)[/tex]
- H: [tex]\(4x^3\)[/tex] and [tex]\(-2x^3\)[/tex]
Thus, Options E, F, and H correctly identify groups of like terms in the polynomial [tex]\(-x^2 + 3x + 4x^3 - 2x^3 + 2x + 7x^2\)[/tex].
First, let's identify all the terms in the polynomial:
- [tex]\(-x^2\)[/tex]
- [tex]\(3x\)[/tex]
- [tex]\(4x^3\)[/tex]
- [tex]\(-2x^3\)[/tex]
- [tex]\(2x\)[/tex]
- [tex]\(7x^2\)[/tex]
Grouping the like terms:
1. Group terms with [tex]\(x^2\)[/tex]:
- [tex]\(-x^2\)[/tex]
- [tex]\(7x^2\)[/tex]
2. Group terms with [tex]\(x\)[/tex]:
- [tex]\(3x\)[/tex]
- [tex]\(2x\)[/tex]
3. Group terms with [tex]\(x^3\)[/tex]:
- [tex]\(4x^3\)[/tex]
- [tex]\(-2x^3\)[/tex]
Next, we will match these groups with the provided options:
- Option A: [tex]\(-x^2\)[/tex] and [tex]\(4x^3\)[/tex]: These are not like terms, because one is [tex]\(x^2\)[/tex] and the other is [tex]\(x^3\)[/tex].
- Option B: [tex]\(3x\)[/tex], [tex]\(7x^2\)[/tex], and [tex]\(-2x^3\)[/tex]: These are not all like terms; each term has a different variable exponent.
- Option C: [tex]\(2x\)[/tex], [tex]\(-x^2\)[/tex], and [tex]\(4x^3\)[/tex]: These are not like terms for the same reason as Option B.
- Option D: [tex]\(7x^2\)[/tex] and [tex]\(-2x^3\)[/tex]: These are not like terms because they have different exponents.
- Option E: [tex]\(-x^2\)[/tex] and [tex]\(7x^2\)[/tex]: These are like terms because they both have [tex]\(x^2\)[/tex].
- Option F: [tex]\(2x\)[/tex] and [tex]\(3x\)[/tex]: These are like terms because they both have [tex]\(x\)[/tex].
- Option G: [tex]\(2x\)[/tex], [tex]\(3x\)[/tex], [tex]\(-x^2\)[/tex], and [tex]\(7x^2\)[/tex]: This option contains two sets of like terms: [tex]\(2x\)[/tex] and [tex]\(3x\)[/tex]; [tex]\(-x^2\)[/tex] and [tex]\(7x^2\)[/tex].
- Option H: [tex]\(4x^3\)[/tex] and [tex]\(-2x^3\)[/tex]: These are like terms because they both have [tex]\(x^3\)[/tex].
After this careful grouping and analysis, we can see that the correct answers are:
- E: [tex]\(-x^2\)[/tex] and [tex]\(7x^2\)[/tex]
- F: [tex]\(2x\)[/tex] and [tex]\(3x\)[/tex]
- H: [tex]\(4x^3\)[/tex] and [tex]\(-2x^3\)[/tex]
Thus, Options E, F, and H correctly identify groups of like terms in the polynomial [tex]\(-x^2 + 3x + 4x^3 - 2x^3 + 2x + 7x^2\)[/tex].
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