To add the given polynomials [tex]\((-5n^4 + 8)\)[/tex] and [tex]\((5n^4 + 8n^3 + 3n^2)\)[/tex], follow these steps:
1. Write down each polynomial:
[tex]\[
(-5n^4 + 8) \quad \text{and} \quad (5n^4 + 8n^3 + 3n^2)
\][/tex]
2. Align the like terms:
Ensure that terms with the same degree of [tex]\(n\)[/tex] are aligned to see clearly which terms can be added directly. In this case:
[tex]\[
-5n^4 + 8
\][/tex]
[tex]\[
5n^4 + 8n^3 + 3n^2
\][/tex]
3. Add the corresponding like terms:
- The [tex]\(n^4\)[/tex] terms: [tex]\((-5n^4 + 5n^4)\)[/tex]
- The [tex]\(n^3\)[/tex] term is only in one polynomial: [tex]\(8n^3\)[/tex]
- The [tex]\(n^2\)[/tex] term is only in one polynomial: [tex]\(3n^2\)[/tex]
- The constant term: [tex]\(8\)[/tex]
4. Combine the like terms:
- For the [tex]\(n^4\)[/tex] terms: [tex]\(-5n^4 + 5n^4 = 0\)[/tex]
- For the [tex]\(n^3\)[/tex] term: [tex]\(8n^3\)[/tex]
- For the [tex]\(n^2\)[/tex] term: [tex]\(3n^2\)[/tex]
- For the constant term: [tex]\(8\)[/tex]
5. Write the final expanded polynomial:
Since the [tex]\(n^4\)[/tex] terms cancel each other out, we are left with:
[tex]\[
8n^3 + 3n^2 + 8
\][/tex]
Thus, the sum of the polynomials is:
[tex]\[
\left(-5n^4 + 8\right) + \left(5n^4 + 8n^3 + 3n^2\right) = 8n^3 + 3n^2 + 8
\][/tex]
