To organize the polynomial expressions based on their degree from least to greatest, follow these steps:
1. Determine the degree of each polynomial expression:
- Expression I: [tex]\(6x^2\)[/tex]
- The highest power of the variable [tex]\(x\)[/tex] is 2.
- Therefore, the degree is 2.
- Expression II: [tex]\(18x^3 + 5ab - 6y\)[/tex]
- The term [tex]\(18x^3\)[/tex] has the highest power of the variable [tex]\(x\)[/tex], which is 3.
- Therefore, the degree is 3.
- Expression III: [tex]\(8a - 5\)[/tex]
- The highest power of the variable [tex]\(a\)[/tex] is 1 (since [tex]\(8a\)[/tex] is essentially [tex]\(8a^1\)[/tex]).
- Therefore, the degree is 1.
- Expression IV: [tex]\(4x^3y + 3x^2 - xy - 4\)[/tex]
- The highest power of the variables when combined is from the term [tex]\(4x^3y\)[/tex]. The combined degree is [tex]\(3 (from x) + 1 (from y) = 4\)[/tex].
- Therefore, the degree is 4.
2. List the degrees of each expression:
- Expression I: Degree 2
- Expression II: Degree 3
- Expression III: Degree 1
- Expression IV: Degree 4
3. Organize the expressions based on their degrees from least to greatest:
- Expression III: Degree 1
- Expression I: Degree 2
- Expression II: Degree 3
- Expression IV: Degree 4
4. Match the expressions with the corresponding indices:
- III is the expression with degree 1.
- I is the expression with degree 2.
- II is the expression with degree 3.
- IV is the expression with degree 4.
Hence, when the polynomial expressions are organized from least to greatest degree, the order is:
- III, I, II, IV.
Therefore, the correct option is:
III, I, II, IV
