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Answer :
To add the given polynomials [tex]\(\left(x^7 + 3x^3 - 4x^2\right)\)[/tex] and [tex]\(\left(4x^3 + 2x\right)\)[/tex], follow these steps:
1. Write down each polynomial:
- The first polynomial is: [tex]\(x^7 + 3x^3 - 4x^2\)[/tex].
- The second polynomial is: [tex]\(4x^3 + 2x\)[/tex].
2. Combine like terms:
- [tex]\(x^7\)[/tex] terms: There is only one term [tex]\(x^7\)[/tex] in the first polynomial.
- [tex]\(x^3\)[/tex] terms: Add the coefficients of the [tex]\(x^3\)[/tex] terms from both polynomials:
- [tex]\(3x^3\)[/tex] from the first polynomial
- [tex]\(4x^3\)[/tex] from the second polynomial
- Total: [tex]\(3x^3 + 4x^3 = 7x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: There is [tex]\(-4x^2\)[/tex] only in the first polynomial, and no [tex]\(x^2\)[/tex] term in the second polynomial.
- [tex]\(x\)[/tex] terms: Add the [tex]\(x\)[/tex] term present only in the second polynomial:
- There is [tex]\(0\)[/tex] coefficient of [tex]\(x\)[/tex] in the first polynomial
- [tex]\(2x\)[/tex] from the second polynomial
- Total: [tex]\(2x\)[/tex]
3. Write the resulting polynomial:
The sum of the polynomials, after combining like terms, is:
[tex]\[
x^7 + 7x^3 - 4x^2 + 2x
\][/tex]
This matches option C: [tex]\(x^7 + 7x^3 - 4x^2 + 2x\)[/tex]. Therefore, the correct answer is:
C. [tex]\(x^7 + 7x^3 - 4x^2 + 2x\)[/tex]
1. Write down each polynomial:
- The first polynomial is: [tex]\(x^7 + 3x^3 - 4x^2\)[/tex].
- The second polynomial is: [tex]\(4x^3 + 2x\)[/tex].
2. Combine like terms:
- [tex]\(x^7\)[/tex] terms: There is only one term [tex]\(x^7\)[/tex] in the first polynomial.
- [tex]\(x^3\)[/tex] terms: Add the coefficients of the [tex]\(x^3\)[/tex] terms from both polynomials:
- [tex]\(3x^3\)[/tex] from the first polynomial
- [tex]\(4x^3\)[/tex] from the second polynomial
- Total: [tex]\(3x^3 + 4x^3 = 7x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: There is [tex]\(-4x^2\)[/tex] only in the first polynomial, and no [tex]\(x^2\)[/tex] term in the second polynomial.
- [tex]\(x\)[/tex] terms: Add the [tex]\(x\)[/tex] term present only in the second polynomial:
- There is [tex]\(0\)[/tex] coefficient of [tex]\(x\)[/tex] in the first polynomial
- [tex]\(2x\)[/tex] from the second polynomial
- Total: [tex]\(2x\)[/tex]
3. Write the resulting polynomial:
The sum of the polynomials, after combining like terms, is:
[tex]\[
x^7 + 7x^3 - 4x^2 + 2x
\][/tex]
This matches option C: [tex]\(x^7 + 7x^3 - 4x^2 + 2x\)[/tex]. Therefore, the correct answer is:
C. [tex]\(x^7 + 7x^3 - 4x^2 + 2x\)[/tex]
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