Question

identify all of the following expressions which represent a polynomial.
select all that apply.
a. \\(\sqrt{2}x^5 - 3x^3y + 5xy^2 - 4x\\)
b. \\(4x^2y - 3xy^2\\)
c. \\(2x^3 + 7x^{-1} + 5\\)
d. \\(6p^2q + 7pq^2 - \frac{5}{p} + \frac{3}{p^2}\\)

Explanation:

Step1: Recall polynomial definition

A polynomial in variables is an expression where each term has non-negative integer exponents on the variables, and no variables in denominators.

Step2: Analyze Option A

Check exponents: All variables have non-negative integer exponents ($x^5, x^3y, xy^2, x^1$), coefficients can be irrational. This is a polynomial.

Step3: Analyze Option B

Check exponents: All variables have non-negative integer exponents ($x^2y, xy^2$). This is a polynomial.

Step4: Analyze Option C

Check exponents: Term $7x^{-1} = \frac{7}{x}$ has a negative exponent. This is not a polynomial.

Step5: Analyze Option D

Check terms: $\frac{5}{p}=5p^{-1}$ and $\frac{3}{p^2}=3p^{-2}$ have negative exponents. This is not a polynomial.

Answer:

A. $\sqrt{2}x^{5}-3x^{3}y + 5xy^{2}-4x$, B. $4x^{2}y - 3xy^{2}$