To determine why the given expression is not a polynomial, we need to carefully analyze each term in the expression:
[tex]\[
\frac{33}{16}-62 y^2 x y-35 z^{\frac{1}{3}} y^2
\][/tex]
Polynomials are algebraic expressions that include terms which are sums or differences of variables raised to non-negative integer exponents, possibly multiplied by coefficients. These coefficients can be real numbers. Let's examine the given expression term by term:
1. [tex]\(\frac{33}{16}\)[/tex]:
- This term is a constant since it does not contain any variables. Constants are allowable in polynomials.
2. [tex]\(-62 y^2 x y\)[/tex]:
- First, combine the variables: [tex]\(y^2 \cdot y = y^3\)[/tex]. Thus, the term becomes [tex]\(-62 y^3 x\)[/tex].
- This term consists of the variables [tex]\(y\)[/tex] raised to the power of 3 and [tex]\(x\)[/tex] raised to the power of 1 (implicitly). Both exponents are non-negative integers, so this term meets the criteria for being part of a polynomial.
3. [tex]\(-35 z^{\frac{1}{3}} y^2\)[/tex]:
- In this term, [tex]\(z\)[/tex] is raised to the power [tex]\(\frac{1}{3}\)[/tex].
- For an expression to be considered a polynomial, all variables in the terms must be raised to non-negative integer exponents. However, [tex]\(\frac{1}{3}\)[/tex] is a fraction rather than an integer.
Given these observations, we focus on the term [tex]\(-35 z^{\frac{1}{3}} y^2\)[/tex]. The fact that this term has a variable ([tex]\(z\)[/tex]) raised to a fractional exponent ([tex]\(\frac{1}{3}\)[/tex]) disqualifies the entire expression from being considered a polynomial.
Hence, the statement that best demonstrates why the given expression is not a polynomial is:
The expression has a variable raised to a fraction.
This correctly explains why the expression does not meet the criteria for being a polynomial. The best reason why the given expression is not a polynomial is because it contains a variable raised to a fractional power.
