To solve this problem, we need to understand the definitions and characteristics of the expressions described in the question.
1. Monomial: A monomial is an algebraic expression that consists of only one term. This means it should not contain any addition or subtraction.
2. 2nd Degree (Quadratic): An expression of the 2nd degree has the highest power of the variable as 2 (e.g., [tex]\( n^2 \)[/tex]).
3. Leading Coefficient: The leading coefficient is the constant factor that multiplies the variable with the highest degree in the expression.
Now, let's analyze each option step by step to see which one fits all these criteria:
### Option (A) [tex]\( 3n^2 \)[/tex]
- Monomial: Yes, this is a single term.
- 2nd Degree: Yes, the highest power of [tex]\( n \)[/tex] is 2.
- Leading Coefficient: Yes, the leading coefficient is 3.
### Option (B) [tex]\( 3n - n^2 \)[/tex]
- Monomial: No, this expression has two terms (it involves subtraction).
- 2nd Degree: It does have a 2nd degree term [tex]\(- n^2 \)[/tex], but it is not a monomial.
- Leading Coefficient: The term [tex]\( -n^2 \)[/tex] indeed has a leading coefficient of -1, but as a whole, it does not match the desired characteristics because it is not a monomial.
### Option (C) [tex]\( 3n^2 - 1 \)[/tex]
- Monomial: No, this expression has two terms (it involves subtraction).
- 2nd Degree: Yes, the term [tex]\( 3n^2 \)[/tex] is of the 2nd degree.
- Leading Coefficient: The term [tex]\( 3n^2 \)[/tex] has a leading coefficient of 3, but the expression contains an additional term, making it not a monomial.
### Option (D) [tex]\( 2n^3 \)[/tex]
- Monomial: Yes, this is a single term.
- 2nd Degree: No, the highest power of [tex]\( n \)[/tex] is 3, not 2.
- Leading Coefficient: The leading coefficient is 2, but it does not fit the requirement of being a 2nd degree expression.
Based on the above analysis, the expression that is a monomial of the 2nd degree with a leading coefficient of 3 is:
(A) [tex]\( 3n^2 \)[/tex]
