To determine why the given expression is a non-example of a polynomial, let's examine the expression step by step:
[tex]\[
\frac{33 y^2}{x^2}-62 y^2 x z-35 z^2 y^2
\][/tex]
Polynomials are algebraic expressions that consist of variables and coefficients, with variables raised to whole number exponents, and no variables in the denominator.
1. First term: [tex]\(\frac{33 y^2}{x^2}\)[/tex]
- This term has a variable [tex]\(x\)[/tex] in the denominator, making it a rational function rather than a polynomial. Polynomials cannot have variables in the denominator of any of their terms.
2. Second term: [tex]\(-62 y^2 x z\)[/tex]
- This term does not have any variables in the denominator. All variables are raised to whole number exponents, which is consistent with polynomial properties.
3. Third term: [tex]\(-35 z^2 y^2\)[/tex]
- Similar to the second term, this term also follows the polynomial definition as it does not have variables in the denominator and all exponents are whole numbers.
Given this analysis, the first term [tex]\(\frac{33 y^2}{x^2}\)[/tex] clearly violates the rule that polynomials cannot have variables in the denominator. This makes the overall expression a non-example of a polynomial.
Therefore, the statement that best demonstrates why the expression is a non-example of a polynomial is:
[tex]\[
\text{The expression has a variable in the denominator of a fraction.}
\][/tex]
This statement correctly identifies the key issue with the given expression, aligning with our analysis.
