Answer :
To add the polynomials [tex]\((7 x^6 + 10 x^2 - 10)\)[/tex] and [tex]\((3 x^6 - 6 x^3 + 4)\)[/tex], follow these steps:
1. Combine like terms:
- Identify and combine the coefficients of terms with the same degree of [tex]\(x\)[/tex].
[tex]\[
(7 x^6 + 10 x^2 - 10) + (3 x^6 - 6 x^3 + 4)
\][/tex]
2. Combine the [tex]\(x^6\)[/tex] terms:
- The first polynomial has [tex]\(7 x^6\)[/tex] and the second has [tex]\(3 x^6\)[/tex].
- Adding these together:
[tex]\[
7 x^6 + 3 x^6 = 10 x^6
\][/tex]
3. Combine the [tex]\(x^3\)[/tex] terms:
- The first polynomial has no [tex]\(x^3\)[/tex] term, and the second has [tex]\(-6 x^3\)[/tex].
- Adding these together:
[tex]\[
0 x^3 - 6 x^3 = -6 x^3
\][/tex]
4. Combine the [tex]\(x^2\)[/tex] terms:
- The first polynomial has [tex]\(10 x^2\)[/tex] and the second has no [tex]\(x^2\)[/tex] term.
- Adding these together:
[tex]\[
10 x^2 + 0 x^2 = 10 x^2
\][/tex]
5. Combine the constant terms:
- The first polynomial has [tex]\(-10\)[/tex] and the second has [tex]\(4\)[/tex].
- Adding these together:
[tex]\[
-10 + 4 = -6
\][/tex]
6. Write down the resulting polynomial by combining all the like terms we identified:
[tex]\[
10 x^6 - 6 x^3 + 10 x^2 - 6
\][/tex]
So, the sum of the polynomials is:
[tex]\[
10 x^6 - 6 x^3 + 10 x^2 - 6
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{10 x^6 - 6 x^3 + 10 x^2 - 6}
\][/tex]
1. Combine like terms:
- Identify and combine the coefficients of terms with the same degree of [tex]\(x\)[/tex].
[tex]\[
(7 x^6 + 10 x^2 - 10) + (3 x^6 - 6 x^3 + 4)
\][/tex]
2. Combine the [tex]\(x^6\)[/tex] terms:
- The first polynomial has [tex]\(7 x^6\)[/tex] and the second has [tex]\(3 x^6\)[/tex].
- Adding these together:
[tex]\[
7 x^6 + 3 x^6 = 10 x^6
\][/tex]
3. Combine the [tex]\(x^3\)[/tex] terms:
- The first polynomial has no [tex]\(x^3\)[/tex] term, and the second has [tex]\(-6 x^3\)[/tex].
- Adding these together:
[tex]\[
0 x^3 - 6 x^3 = -6 x^3
\][/tex]
4. Combine the [tex]\(x^2\)[/tex] terms:
- The first polynomial has [tex]\(10 x^2\)[/tex] and the second has no [tex]\(x^2\)[/tex] term.
- Adding these together:
[tex]\[
10 x^2 + 0 x^2 = 10 x^2
\][/tex]
5. Combine the constant terms:
- The first polynomial has [tex]\(-10\)[/tex] and the second has [tex]\(4\)[/tex].
- Adding these together:
[tex]\[
-10 + 4 = -6
\][/tex]
6. Write down the resulting polynomial by combining all the like terms we identified:
[tex]\[
10 x^6 - 6 x^3 + 10 x^2 - 6
\][/tex]
So, the sum of the polynomials is:
[tex]\[
10 x^6 - 6 x^3 + 10 x^2 - 6
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{10 x^6 - 6 x^3 + 10 x^2 - 6}
\][/tex]
