Answer :
To solve the problem of adding the two polynomials [tex]\((7x^6 + 10x^2 - 10)\)[/tex] and [tex]\((3x^6 - 6x^3 + 4)\)[/tex], we simply need to add the coefficients of terms with the same degree. Let's break it down step by step:
1. Identify Like Terms:
- Degrees of polynomials in each expression:
- [tex]\(7x^6\)[/tex] and [tex]\(3x^6\)[/tex] both have [tex]\(x^6\)[/tex].
- There is no [tex]\(x^3\)[/tex] term in the first polynomial, but there is [tex]\(-6x^3\)[/tex] in the second polynomial.
- [tex]\(10x^2\)[/tex] and there is no [tex]\(x^2\)[/tex] term in the second polynomial.
- Constant terms: [tex]\(-10\)[/tex] (first polynomial) and [tex]\(4\)[/tex] (second polynomial).
2. Add the Coefficients:
- For the [tex]\(x^6\)[/tex] terms:
[tex]\[
7 + 3 = 10
\][/tex]
- For the [tex]\(x^3\)[/tex] terms (only present in the second polynomial):
[tex]\[
0 - 6 = -6
\][/tex]
- For the [tex]\(x^2\)[/tex] terms:
[tex]\[
10 + 0 = 10
\][/tex]
- For the constant terms:
[tex]\[
-10 + 4 = -6
\][/tex]
3. Write Down the Result:
- Combine the results to form the new polynomial:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]
Thus, the final polynomial after addition is:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]
The correct option from the given choices is:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]
1. Identify Like Terms:
- Degrees of polynomials in each expression:
- [tex]\(7x^6\)[/tex] and [tex]\(3x^6\)[/tex] both have [tex]\(x^6\)[/tex].
- There is no [tex]\(x^3\)[/tex] term in the first polynomial, but there is [tex]\(-6x^3\)[/tex] in the second polynomial.
- [tex]\(10x^2\)[/tex] and there is no [tex]\(x^2\)[/tex] term in the second polynomial.
- Constant terms: [tex]\(-10\)[/tex] (first polynomial) and [tex]\(4\)[/tex] (second polynomial).
2. Add the Coefficients:
- For the [tex]\(x^6\)[/tex] terms:
[tex]\[
7 + 3 = 10
\][/tex]
- For the [tex]\(x^3\)[/tex] terms (only present in the second polynomial):
[tex]\[
0 - 6 = -6
\][/tex]
- For the [tex]\(x^2\)[/tex] terms:
[tex]\[
10 + 0 = 10
\][/tex]
- For the constant terms:
[tex]\[
-10 + 4 = -6
\][/tex]
3. Write Down the Result:
- Combine the results to form the new polynomial:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]
Thus, the final polynomial after addition is:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]
The correct option from the given choices is:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]
