To determine which algebraic expression is a polynomial of degree 3, we need to recognize the characteristics of polynomials. A polynomial expression consists of terms in which variables are raised to non-negative integer powers and the coefficients are real numbers. The degree of a polynomial is the highest power of the variable in the expression.
Let's analyze each expression one by one:
1. [tex]\(4 x^3 - \frac{2}{x}\)[/tex]
- The term [tex]\(4 x^3\)[/tex] has a variable [tex]\(x\)[/tex] raised to the power of 3.
- The term [tex]\(-\frac{2}{x}\)[/tex] can be rewritten as [tex]\(-2 x^{-1}\)[/tex], which involves a negative power of [tex]\(x\)[/tex].
Since having a variable in the denominator or with a negative power disqualifies the expression from being a polynomial, this expression is not a polynomial.
2. [tex]\(2 y^3 + 5 y^2 - 5 y\)[/tex]
- The term [tex]\(2 y^3\)[/tex] has a variable [tex]\(y\)[/tex] raised to the power of 3.
- The term [tex]\(5 y^2\)[/tex] has a variable [tex]\(y\)[/tex] raised to the power of 2.
- The term [tex]\(-5 y\)[/tex] has a variable [tex]\(y\)[/tex] raised to the power of 1.
All terms have positive integer exponents and are sum of variable terms. Hence, this expression is a polynomial. The highest degree term is [tex]\(2 y^3\)[/tex], which is of degree 3.
3. [tex]\(3 y^3 - \sqrt{4 y}\)[/tex]
- The term [tex]\(3 y^3\)[/tex] has a variable [tex]\(y\)[/tex] raised to the power of 3.
- The term [tex]\(\sqrt{4 y}\)[/tex] can be rewritten as [tex]\(2 \sqrt{y}\)[/tex], which involves a variable [tex]\(y\)[/tex] raised to the power of [tex]\( \frac{1}{2} \)[/tex].
Since the expression includes a term with a fractional exponent, it is not a polynomial.
4. [tex]\(-x y \sqrt{6}\)[/tex]
- The term [tex]\(-x y \sqrt{6}\)[/tex] can be considered as [tex]\(- \sqrt{6} xy\)[/tex].
The term involves the product of variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex], meaning variables have degrees 1 each. However, the overall term does not conform to the polynomial form in terms of a single variable's positive integer exponents within each term, thus making it not a polynomial.
Among the given expressions, the only one that qualifies as a polynomial of degree 3 is:
[tex]\[2 y^3 + 5 y^2 - 5 y\][/tex]
So, the correct answer is the second expression:
[tex]\[2 y^3 + 5 y^2 - 5 y\][/tex]
