To solve this problem, we need to identify the expression that matches the given description: a 3rd degree binomial with a constant term of 8. Let's review each option carefully.
1. Option (A) [tex]\(2x^8 + 3\)[/tex]:
- The degree of this polynomial is 8 because the highest power of [tex]\(x\)[/tex] is [tex]\(x^8\)[/tex].
- It has more than two terms, so it is not a binomial.
- It does not match the required condition of being a 3rd degree binomial.
2. Option (B) [tex]\(8x^3 + 2x + 3\)[/tex]:
- The degree of this polynomial is 3 because the highest power of [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].
- It has three terms: [tex]\(8x^3\)[/tex], [tex]\(2x\)[/tex], and 3. This means it is not a binomial (a binomial should have exactly two terms).
- Hence, it does not match the condition of having exactly two terms.
3. Option (C) [tex]\(x^3 - x^2 + 8\)[/tex]:
- The degree of this polynomial is 3, as the highest power of [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].
- It has three terms: [tex]\(x^3\)[/tex], [tex]\(-x^2\)[/tex], and 8, making it a trinomial, not a binomial.
- Therefore, it does not meet the criterion of being a binomial.
4. Option (D) [tex]\(-5x^3 + 8\)[/tex]:
- The degree of this polynomial is 3 because the highest power of [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].
- It has exactly two terms: [tex]\(-5x^3\)[/tex] and 8, making it a binomial.
- Additionally, it has a constant term of 8, which matches the given description.
Based on the analysis, the correct choice is:
(D) [tex]\(-5x^3 + 8\)[/tex].
