To organize the given polynomial expressions based on their degree from least to greatest, let's first determine the degree of each expression.
### Polynomial I: [tex]\( x + 2xy^1z^1 \)[/tex]
- Degree of [tex]\( x \)[/tex] is 1.
- Degree of [tex]\( 2xyz \)[/tex] is [tex]\( 1 (x) + 1 (y) + 1 (z) = 3 \)[/tex].
Since the degree of a polynomial is the highest degree of any term in it, the degree of Polynomial I is [tex]\( 3 \)[/tex].
### Polynomial II: [tex]\( 9x^3y^2 \)[/tex]
- There is only one term, [tex]\( 9x^3y^2 \)[/tex].
- Degree of [tex]\( 9x^3y^2 \)[/tex] is [tex]\( 3 (x) + 2 (y) = 5 \)[/tex].
So, the degree of Polynomial II is [tex]\( 5 \)[/tex].
### Polynomial III: [tex]\( 18x^2 + 5ab - 6y \)[/tex]
- Degree of [tex]\( 18x^2 \)[/tex] is 2.
- Degree of [tex]\( 5ab \)[/tex] is [tex]\( 0 (a and b are constants) \)[/tex].
- Degree of [tex]\( -6y \)[/tex] is 1.
The highest degree term in Polynomial III is [tex]\( 18x^2 \)[/tex], which has a degree of [tex]\( 2 \)[/tex].
### Polynomial IV: [tex]\( 4x^4 + 3x^2 - x - 4 \)[/tex]
- Degree of [tex]\( 4x^4 \)[/tex] is 4.
- Degree of [tex]\( 3x^2 \)[/tex] is 2.
- Degree of [tex]\( -x \)[/tex] is 1.
- Degree of [tex]\( -4 \)[/tex] is 0.
The highest degree term in Polynomial IV is [tex]\( 4x^4 \)[/tex], which has a degree of [tex]\( 4 \)[/tex].
Now, let's list the degrees:
- Polynomial I: 3
- Polynomial II: 5
- Polynomial III: 2
- Polynomial IV: 4
Organizing these from least to greatest:
1. Polynomial III (degree 2)
2. Polynomial I (degree 3)
3. Polynomial IV (degree 4)
4. Polynomial II (degree 5)
Thus, the correct order of the polynomial expressions from least to greatest based on their degree is:
III, I, IV, II.
