To classify each polynomial and determine its degree, let's break down the polynomial [tex]\( x^2 y + 3 x y^2 + 1 \)[/tex].
### Step-by-Step Solution:
1. Classification:
- A polynomial with one term is called a monomial.
- A polynomial with two terms is called a binomial.
- A polynomial with three terms is called a trinomial.
The given polynomial [tex]\( x^2 y + 3 x y^2 + 1 \)[/tex] has three terms: [tex]\( x^2 y \)[/tex], [tex]\( 3 x y^2 \)[/tex], and [tex]\( 1 \)[/tex]. Therefore, it is a trinomial.
2. Determining the Degree:
- The degree of a term is the sum of the exponents of the variables in that term.
- The degree of the polynomial is the highest degree of its terms.
Now let's find the degree of each term:
- Term 1: [tex]\( x^2 y \)[/tex]
- The degree is [tex]\( 2 \)[/tex] (from [tex]\( x^2 \)[/tex]) + [tex]\( 1 \)[/tex] (from [tex]\( y \)[/tex]) = [tex]\( 3 \)[/tex].
- Term 2: [tex]\( 3 x y^2 \)[/tex]
- The degree is [tex]\( 1 \)[/tex] (from [tex]\( x \)[/tex]) + [tex]\( 2 \)[/tex] (from [tex]\( y^2 \)[/tex]) = [tex]\( 3 \)[/tex].
- Term 3: [tex]\( 1 \)[/tex]
- This is a constant term with no variables, so its degree is [tex]\( 0 \)[/tex].
The highest degree among these terms is [tex]\( 3 \)[/tex].
### Final Answer:
- The polynomial [tex]\( x^2 y + 3 x y^2 + 1 \)[/tex] is a trinomial with a degree of 3.
So, the completed statement would be:
>The polynomial [tex]\( x^2 y + 3 x y^2 + 1 \)[/tex] is a trinomial with a degree of 3.
