To solve the given problem, we'll classify each polynomial and determine the degree of each polynomial.
### 1. Polynomial: [tex]\(3x^2\)[/tex]
First, let's classify the polynomial [tex]\(3x^2\)[/tex]:
- A monomial is a polynomial with exactly one term.
- Since [tex]\(3x^2\)[/tex] has only one term, it is classified as a monomial.
Next, let's determine the degree of the polynomial [tex]\(3x^2\)[/tex]:
- The degree of a monomial is the highest power of its variable. Here, the term is [tex]\(3x^2\)[/tex].
- The highest power of [tex]\(x\)[/tex] in this term is [tex]\(2\)[/tex].
Thus, the polynomial [tex]\(3x^2\)[/tex] is a monomial with a degree of [tex]\(2\)[/tex].
### 2. Polynomial: [tex]\(x^2y + 3xy^2 + 1\)[/tex]
Next, let's classify the polynomial [tex]\(x^2y + 3xy^2 + 1\)[/tex]:
- A trinomial is a polynomial with exactly three terms.
- Since [tex]\(x^2y + 3xy^2 + 1\)[/tex] has three terms ([tex]\(x^2y\)[/tex], [tex]\(3xy^2\)[/tex], and [tex]\(1\)[/tex]), it is classified as a trinomial.
Now, let's determine the degree of the polynomial [tex]\(x^2y + 3xy^2 + 1\)[/tex]:
- The degree of a polynomial is determined by the term with the highest sum of exponents of the variables.
- Term [tex]\(x^2y\)[/tex]: The sum of exponents is [tex]\(2 + 1 = 3\)[/tex].
- Term [tex]\(3xy^2\)[/tex]: The sum of exponents is [tex]\(1 + 2 = 3\)[/tex].
- Term [tex]\(1\)[/tex]: The sum of exponents is [tex]\(0\)[/tex].
The highest sum of exponents among these terms is [tex]\(3\)[/tex].
Thus, the polynomial [tex]\(x^2y + 3xy^2 + 1\)[/tex] is a trinomial with a degree of [tex]\(3\)[/tex].
Summarizing:
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.
