To determine which terms from the given set can be the first term in a polynomial with a degree of 5 written in standard form, we need to check the degree of each term. The degree of a term in a polynomial is the sum of the exponents of all the variables in that term.
Let’s analyze each term:
1. [tex]\(-4x^3y^2\)[/tex]
- Degree of [tex]\(x\)[/tex] = 3
- Degree of [tex]\(y\)[/tex] = 2
- Total degree = 3 (from [tex]\(x\)[/tex]) + 2 (from [tex]\(y\)[/tex]) = 5
- This term has a degree of 5.
2. [tex]\(x^3\)[/tex]
- Degree of [tex]\(x\)[/tex] = 3
- There is no [tex]\(y\)[/tex] term.
- Total degree = 3
- This term has a degree of 3, which is not equal to 5.
3. [tex]\(8.4x^4y^2\)[/tex]
- Degree of [tex]\(x\)[/tex] = 4
- Degree of [tex]\(y\)[/tex] = 2
- Total degree = 4 (from [tex]\(x\)[/tex]) + 2 (from [tex]\(y\)[/tex]) = 6
- This term has a degree of 6, which is not equal to 5.
4. [tex]\(5x^4y\)[/tex]
- Degree of [tex]\(x\)[/tex] = 4
- Degree of [tex]\(y\)[/tex] = 1
- Total degree = 4 (from [tex]\(x\)[/tex]) + 1 (from [tex]\(y\)[/tex]) = 5
- This term has a degree of 5.
5. [tex]\(-xy^3\)[/tex]
- Degree of [tex]\(x\)[/tex] = 1
- Degree of [tex]\(y\)[/tex] = 3
- Total degree = 1 (from [tex]\(x\)[/tex]) + 3 (from [tex]\(y\)[/tex]) = 4
- This term has a degree of 4, which is not equal to 5.
6. [tex]\(\frac{-2x^4}{y}\)[/tex]
- We need to rewrite this term to see the degrees clearly: [tex]\(-2x^4y^{-1}\)[/tex]
- Degree of [tex]\(x\)[/tex] = 4
- Degree of [tex]\(y\)[/tex] = -1
- Total degree = 4 (from [tex]\(x\)[/tex]) + (-1) (from [tex]\(y\)[/tex]) = 3
- This term has a degree of 3, which is not equal to 5.
After evaluating all the terms, we find that the terms that have a degree of 5 are:
- [tex]\(-4x^3y^2\)[/tex]
- [tex]\(5x^4y\)[/tex]
Therefore, these two terms can be used as the first term to create a polynomial with a degree of 5 written in standard form.
