To analyze the polynomial [tex]\( 9x^2y - 6x - 5y^2 \)[/tex], we need to determine both its classification by the number of terms and its degree.
### Step 1: Counting the Number of Terms
A polynomial's classification depends on the number of terms:
- A binomial has 2 terms.
- A trinomial has 3 terms.
The given polynomial [tex]\( 9x^2y - 6x - 5y^2 \)[/tex] consists of three distinct terms:
1. [tex]\( 9x^2y \)[/tex]
2. [tex]\( -6x \)[/tex]
3. [tex]\( -5y^2 \)[/tex]
Since there are three terms, this polynomial is classified as a trinomial.
### Step 2: Determining the Degree
The degree of a polynomial is determined by the highest sum of the exponents of its variables in any single term.
Let's analyze the degree of each term in the polynomial:
1. [tex]\( 9x^2y \)[/tex]:
- The term [tex]\( 9x^2y \)[/tex] has variables with exponents [tex]\( x^2 \)[/tex] (where the exponent is 2) and [tex]\( y \)[/tex] (where the exponent is 1).
- The total degree of this term is [tex]\( 2 + 1 = 3 \)[/tex].
2. [tex]\( -6x \)[/tex]:
- The term [tex]\( -6x \)[/tex] has a variable [tex]\( x \)[/tex] with an exponent of 1.
- The total degree of this term is [tex]\( 1 \)[/tex].
3. [tex]\( -5y^2 \)[/tex]:
- The term [tex]\( -5y^2 \)[/tex] has a variable [tex]\( y \)[/tex] with an exponent of 2.
- The total degree of this term is [tex]\( 2 \)[/tex].
The highest degree among the terms is 3.
### Conclusion
The polynomial [tex]\( 9x^2y - 6x - 5y^2 \)[/tex] is a trinomial with a degree of 3.
Thus, the correct statement about the polynomial is:
- It is a trinomial with a degree of 3.
