College

Add \([7x^6 + 10x^2 - 10] + [3x^6 - 6x^3 + 4]\).

A. \(10x^6 + 4x^2 - 6\)

B. \(10x^6 + 4x^3 - 6\)

C. \(10x^6 - 6x^3 + 10x^2 - 6\)

D. \(10x^6 + 10x^2 - 6x^3 + 6\)

Answer :

To add the two given polynomials [tex]\((7x^6 + 10x^2 - 10)\)[/tex] and [tex]\((3x^6 - 6x^3 + 4)\)[/tex], follow these steps:

1. Identify and Align Like Terms:
- The first polynomial is [tex]\(7x^6 + 0x^5 + 10x^2 + 0x^1 - 10\)[/tex].
- The second polynomial is [tex]\(3x^6 + 0x^5 - 6x^3 + 0x^2 + 0x^1 + 4\)[/tex].
- Align the terms according to their powers: [tex]\(x^6, x^5, x^4, x^3, x^2, x^1,\)[/tex] and the constant term.

2. Add Coefficients of Like Terms:
- For [tex]\(x^6\)[/tex]: [tex]\(7 + 3 = 10\)[/tex]
- For [tex]\(x^5\)[/tex]: [tex]\(0 + 0 = 0\)[/tex]
- For [tex]\(x^4\)[/tex]: No term, so [tex]\(0 + 0 = 0\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(0 - 6 = -6\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(10 + 0 = 10\)[/tex]
- For [tex]\(x^1\)[/tex]: [tex]\(0 + 0 = 0\)[/tex]
- For the constant term: [tex]\(-10 + 4 = -6\)[/tex]

3. Write the Resulting Polynomial:
- Combine these coefficients into a single polynomial expression: [tex]\(10x^6 - 6x^3 + 10x^2 - 6\)[/tex].

Therefore, the result of adding the two polynomials is [tex]\(10x^6 - 6x^3 + 10x^2 - 6\)[/tex], which matches the choice:

[tex]\[10x^6 - 6x^3 + 10x^2 - 6\][/tex]

This matches the option:
[tex]\[10x^6 - 6x^3 + 10x^2 - 6\][/tex]

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