Absolutely, let's analyze each polynomial step-by-step to classify and determine its degree.
### Polynomial: [tex]\( 3x^2 \)[/tex]
1. Form: This polynomial consists of a single term, [tex]\( 3x^2 \)[/tex].
2. Type: A polynomial with only one term is called a monomial.
3. Degree: The degree of a monomial is simply the exponent of the variable. Here the highest exponent of [tex]\(x\)[/tex] is 2.
Therefore, [tex]\( 3x^2 \)[/tex] is a monomial with a degree of 2.
### Polynomial: [tex]\( x^2y + 3xy^2 + 1 \)[/tex]
1. Form: This polynomial consists of three terms: [tex]\( x^2y \)[/tex], [tex]\( 3xy^2 \)[/tex], and 1.
2. Type: A polynomial with three terms is called a trinomial.
3. Degree: For polynomials with multiple variables, the degree of a term is the sum of the exponents of all variables within that term. Let's examine the degree of each term:
- [tex]\( x^2y \)[/tex]: The sum of the exponents is [tex]\( 2 + 1 = 3 \)[/tex].
- [tex]\( 3xy^2 \)[/tex]: The sum of the exponents is [tex]\( 1 + 2 = 3 \)[/tex].
- 1: This is a constant term with a degree of 0.
The degree of the polynomial is the highest degree of its terms. Here, both [tex]\( x^2y \)[/tex] and [tex]\( 3xy^2 \)[/tex] have the highest degree, which is 3.
Therefore, [tex]\( x^2y + 3xy^2 + 1 \)[/tex] is a trinomial with a degree of 3.
### Final Answer:
- The polynomial [tex]\( 3x^2 \)[/tex] is a monomial with a degree of 2.
- The polynomial [tex]\( x^2y + 3xy^2 + 1 \)[/tex] is a trinomial with a degree of 3.
