High School

Add the polynomials:

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

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], let's go through it step-by-step:

1. Identify and Combine Like Terms:

- Terms involving [tex]\(x^6\)[/tex]:
- From the first polynomial: [tex]\(7x^6\)[/tex]
- From the second polynomial: [tex]\(3x^6\)[/tex]
- Combine them: [tex]\(7x^6 + 3x^6 = 10x^6\)[/tex]

- Terms involving [tex]\(x^3\)[/tex]:
- From the first polynomial: There is no [tex]\(x^3\)[/tex] term (consider it as [tex]\(0x^3\)[/tex])
- From the second polynomial: [tex]\(-6x^3\)[/tex]
- Combine them: [tex]\(0x^3 - 6x^3 = -6x^3\)[/tex]

- Terms involving [tex]\(x^2\)[/tex]:
- From the first polynomial: [tex]\(10x^2\)[/tex]
- From the second polynomial: There is no [tex]\(x^2\)[/tex] term (consider it as [tex]\(0x^2\)[/tex])
- Combine them: [tex]\(10x^2 + 0x^2 = 10x^2\)[/tex]

- Constant terms (without x):
- From the first polynomial: [tex]\(-10\)[/tex]
- From the second polynomial: [tex]\(4\)[/tex]
- Combine them: [tex]\(-10 + 4 = -6\)[/tex]

2. Form the Resultant Polynomial:

Putting it all together, the resultant polynomial is:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]

Therefore, the answer is:
[tex]\[ 10x^6 - 6x^3 + 10x^2 - 6 \][/tex]
The correct option that matches this result is:
[tex]\[ 10x^6 - 6x^3 + 10x^2 - 6 \][/tex]

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