To solve the given polynomial expression [tex]\(z^3 - 11z^4 - 5z^2 + z^4 - 5z^3\)[/tex], we need to combine like terms. Let's break it down step by step:
1. Identify like terms:
- [tex]\(z^4\)[/tex] terms: [tex]\(-11z^4\)[/tex] and [tex]\(+z^4\)[/tex]
- [tex]\(z^3\)[/tex] terms: [tex]\(z^3\)[/tex] and [tex]\(-5z^3\)[/tex]
- [tex]\(z^2\)[/tex] terms: [tex]\(-5z^2\)[/tex]
2. Combine the like terms:
- Combine [tex]\(z^4\)[/tex] terms: [tex]\(-11z^4 + z^4 = -10z^4\)[/tex]
- Combine [tex]\(z^3\)[/tex] terms: [tex]\(z^3 - 5z^3 = -4z^3\)[/tex]
- The [tex]\(z^2\)[/tex] term remains as: [tex]\(-5z^2\)[/tex]
3. Write the simplified polynomial:
- The combined polynomial terms are: [tex]\(-10z^4 - 4z^3 - 5z^2\)[/tex]
4. Match the simplified polynomial with given options:
- A. [tex]\(10z^4 + 6z^3 + 5z^2\)[/tex]
- B. [tex]\(-10z^4 - 4z^3 - 5z^2\)[/tex]
- C. [tex]\(-10z^8 - 6z^6 - 5z^2\)[/tex]
- D. [tex]\(-11z^4 - z^3 - 5z^2\)[/tex]
By comparing our combined polynomial [tex]\(-10z^4 - 4z^3 - 5z^2\)[/tex] with the given options, we see that it matches Option B.
Therefore, the correct answer is: B. [tex]\(-10z^4 - 4z^3 - 5z^2\)[/tex].
