Answer :
To add the two polynomials [tex]\((7x^6 + 10x^2 - 10)\)[/tex] and [tex]\((3x^6 - 6x^3 + 4)\)[/tex], follow these steps:
1. Identify Like Terms: Look for terms in each polynomial that have the same variable raised to the same power.
- In the first polynomial, the terms are [tex]\(7x^6\)[/tex], [tex]\(10x^2\)[/tex], and [tex]\(-10\)[/tex].
- In the second polynomial, the terms are [tex]\(3x^6\)[/tex], [tex]\(-6x^3\)[/tex], and [tex]\(4\)[/tex].
2. Add the Coefficients of Like Terms:
- For [tex]\(x^6\)[/tex] terms: Add [tex]\(7x^6\)[/tex] and [tex]\(3x^6\)[/tex]:
[tex]\[
7x^6 + 3x^6 = 10x^6
\][/tex]
- For [tex]\(x^3\)[/tex] terms: Here only the second polynomial has a term [tex]\(-6x^3\)[/tex]. Thus, it remains:
[tex]\[
-6x^3
\][/tex]
- For [tex]\(x^2\)[/tex] terms: Add [tex]\(10x^2\)[/tex] from the first polynomial, and there is no [tex]\(x^2\)[/tex] term in the second polynomial, so:
[tex]\[
10x^2
\][/tex]
- For the constant terms (i.e., the [tex]\(x^0\)[/tex] terms): Add [tex]\(-10\)[/tex] and [tex]\(4\)[/tex]:
[tex]\[
-10 + 4 = -6
\][/tex]
3. Combine All the Terms: Combine the results from each like-term addition:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]
The result of adding the two polynomials is [tex]\(10x^6 - 6x^3 + 10x^2 - 6\)[/tex].
1. Identify Like Terms: Look for terms in each polynomial that have the same variable raised to the same power.
- In the first polynomial, the terms are [tex]\(7x^6\)[/tex], [tex]\(10x^2\)[/tex], and [tex]\(-10\)[/tex].
- In the second polynomial, the terms are [tex]\(3x^6\)[/tex], [tex]\(-6x^3\)[/tex], and [tex]\(4\)[/tex].
2. Add the Coefficients of Like Terms:
- For [tex]\(x^6\)[/tex] terms: Add [tex]\(7x^6\)[/tex] and [tex]\(3x^6\)[/tex]:
[tex]\[
7x^6 + 3x^6 = 10x^6
\][/tex]
- For [tex]\(x^3\)[/tex] terms: Here only the second polynomial has a term [tex]\(-6x^3\)[/tex]. Thus, it remains:
[tex]\[
-6x^3
\][/tex]
- For [tex]\(x^2\)[/tex] terms: Add [tex]\(10x^2\)[/tex] from the first polynomial, and there is no [tex]\(x^2\)[/tex] term in the second polynomial, so:
[tex]\[
10x^2
\][/tex]
- For the constant terms (i.e., the [tex]\(x^0\)[/tex] terms): Add [tex]\(-10\)[/tex] and [tex]\(4\)[/tex]:
[tex]\[
-10 + 4 = -6
\][/tex]
3. Combine All the Terms: Combine the results from each like-term addition:
[tex]\[
10x^6 - 6x^3 + 10x^2 - 6
\][/tex]
The result of adding the two polynomials is [tex]\(10x^6 - 6x^3 + 10x^2 - 6\)[/tex].
