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Answer :
To add the two polynomials [tex]\((3x^3 + 5x^2 - 2) + (4x^3 - 5x^2 - 2x + 6)\)[/tex], follow these steps:
1. Identify the Terms:
- The first polynomial, [tex]\(3x^3 + 5x^2 - 2\)[/tex], contains the terms:
- [tex]\(3x^3\)[/tex] (cubic term)
- [tex]\(5x^2\)[/tex] (quadratic term)
- [tex]\(-2\)[/tex] (constant term)
- The second polynomial, [tex]\(4x^3 - 5x^2 - 2x + 6\)[/tex], contains the terms:
- [tex]\(4x^3\)[/tex] (cubic term)
- [tex]\(-5x^2\)[/tex] (quadratic term)
- [tex]\(-2x\)[/tex] (linear term)
- [tex]\(6\)[/tex] (constant term)
2. Combine Like Terms:
- Cubic Terms: [tex]\(3x^3 + 4x^3 = 7x^3\)[/tex]
- Quadratic Terms: [tex]\(5x^2 - 5x^2 = 0\)[/tex] (these cancel each other out)
- Linear Terms: There is no [tex]\(x\)[/tex] term in the first polynomial, so just bring down [tex]\(-2x\)[/tex].
- Constant Terms: [tex]\(-2 + 6 = 4\)[/tex]
3. Form the Resultant Polynomial:
- Combine all the terms we found: [tex]\(7x^3 + 0x^2 - 2x + 4\)[/tex].
4. Simplify:
- Since the [tex]\(x^2\)[/tex] term is zero, we don't include it in our final expression.
- So, the resulting polynomial is [tex]\(7x^3 - 2x + 4\)[/tex].
Therefore, the resultant polynomial is [tex]\(7x^3 - 2x + 4\)[/tex], which corresponds to option (C).
1. Identify the Terms:
- The first polynomial, [tex]\(3x^3 + 5x^2 - 2\)[/tex], contains the terms:
- [tex]\(3x^3\)[/tex] (cubic term)
- [tex]\(5x^2\)[/tex] (quadratic term)
- [tex]\(-2\)[/tex] (constant term)
- The second polynomial, [tex]\(4x^3 - 5x^2 - 2x + 6\)[/tex], contains the terms:
- [tex]\(4x^3\)[/tex] (cubic term)
- [tex]\(-5x^2\)[/tex] (quadratic term)
- [tex]\(-2x\)[/tex] (linear term)
- [tex]\(6\)[/tex] (constant term)
2. Combine Like Terms:
- Cubic Terms: [tex]\(3x^3 + 4x^3 = 7x^3\)[/tex]
- Quadratic Terms: [tex]\(5x^2 - 5x^2 = 0\)[/tex] (these cancel each other out)
- Linear Terms: There is no [tex]\(x\)[/tex] term in the first polynomial, so just bring down [tex]\(-2x\)[/tex].
- Constant Terms: [tex]\(-2 + 6 = 4\)[/tex]
3. Form the Resultant Polynomial:
- Combine all the terms we found: [tex]\(7x^3 + 0x^2 - 2x + 4\)[/tex].
4. Simplify:
- Since the [tex]\(x^2\)[/tex] term is zero, we don't include it in our final expression.
- So, the resulting polynomial is [tex]\(7x^3 - 2x + 4\)[/tex].
Therefore, the resultant polynomial is [tex]\(7x^3 - 2x + 4\)[/tex], which corresponds to option (C).
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