College

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

Choose the correct answer:

A. [tex]10x^6 + 4x^2 - 6[/tex]

B. [tex]10x^6 + 4x^3 - 6[/tex]

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

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

Answer :

To add the polynomials [tex]\((7x^6 + 10x^2 - 10)\)[/tex] and [tex]\((3x^6 - 6x^3 + 4)\)[/tex], we need to combine like terms. Here’s how you can do it step-by-step:

1. Identify and align the like terms:
- For the first polynomial [tex]\(7x^6 + 10x^2 - 10\)[/tex], the terms are:
- [tex]\(7x^6\)[/tex]
- [tex]\(10x^2\)[/tex]
- [tex]\(-10\)[/tex]

- For the second polynomial [tex]\(3x^6 - 6x^3 + 4\)[/tex], the terms are:
- [tex]\(3x^6\)[/tex]
- [tex]\(-6x^3\)[/tex]
- [tex]\(4\)[/tex]

2. Add the coefficients of like terms:
- [tex]\(x^6\)[/tex] terms: [tex]\(7x^6 + 3x^6 = 10x^6\)[/tex]
- [tex]\(x^3\)[/tex] term: Since there is no [tex]\(x^3\)[/tex] term in the first polynomial, it remains as [tex]\(-6x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(10x^2\)[/tex] (from the first polynomial, there’s no [tex]\(x^2\)[/tex] term in the second polynomial)
- Constant terms: [tex]\(-10 + 4 = -6\)[/tex]

3. Combine the results to form the new polynomial:
- The final result of adding these two polynomials is:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]

So, the correct polynomial after addition is:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]

This matches option [tex]\( \boxed{10x^6 - 6x^3 + 10x^2 - 6} \)[/tex].

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