To organize the given polynomial expressions from least to greatest based on their degree, let’s first determine the degree of each polynomial.
### Step-by-Step Determination of Polynomial Degrees:
1. Polynomial I: [tex]\(x + 2xy\)[/tex]
- [tex]\(x\)[/tex] has a degree of 1.
- [tex]\(2xy\)[/tex] has a degree of [tex]\(1 + 1 = 2\)[/tex].
- The highest degree term is [tex]\(2xy\)[/tex], so the degree of Polynomial I is 2.
2. Polynomial II: [tex]\(3x + y + z\)[/tex]
- [tex]\(3x\)[/tex] has a degree of 1.
- [tex]\(y\)[/tex] has a degree of 1.
- [tex]\(z\)[/tex] has a degree of 1.
- The highest degree term among them is 1, so the degree of Polynomial II is 1.
3. Polynomial III: [tex]\(2x^3y + y^2x - 3x + 4\)[/tex]
- [tex]\(2x^3y\)[/tex] has a degree of [tex]\(3 + 1 = 4\)[/tex].
- [tex]\(y^2x\)[/tex] has a degree of [tex]\(2 + 1 = 3\)[/tex].
- [tex]\(-3x\)[/tex] has a degree of 1.
- [tex]\(4\)[/tex] (constant term) has a degree of 0.
- The highest degree term is [tex]\(2x^3y\)[/tex], which has a degree of 4. Therefore, the degree of Polynomial III is 4.
4. Polynomial IV: [tex]\(9x^3yz\)[/tex]
- [tex]\(9x^3yz\)[/tex] has a degree of [tex]\(3 + 1 + 1 = 5\)[/tex].
- The degree of Polynomial IV is 5.
### Organizing the Polynomials:
Now that we know the degrees of each polynomial:
- Polynomial I: degree 2
- Polynomial II: degree 1
- Polynomial III: degree 4
- Polynomial IV: degree 5
We can list the polynomials in ascending order based on their degrees:
Ascending Order:
1. Polynomial II (degree 1)
2. Polynomial I (degree 2)
3. Polynomial III (degree 4)
4. Polynomial IV (degree 5)
Therefore, the correct order of the polynomials from least to greatest based on their degree is:
II, I, III, IV
Thus, the answer is:
II, I, III, IV
