State whether the given expressions are polynomials or not.
(a) $3 x^{3} - 4 x y + z^{2} - 3 x + 3 y$
(b) $3 a^{2} b - 4 \sqrt{b^{3}} + b^{2}$
(c) $\frac{5}{2} x^{2} - 4 y z - \frac{5}{6} x^{2}$
(d) $3 p^{3} - 4 q^{\frac{2}{3}} + p q r + 3 p q$

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Solution

(a) $3 x^{3} - 4 x y + z^{2} - 3 x + 3 y$
Yes. Because there are no negative or fractional powers for the variables.

(b) $3 a^{2} b - 4 \sqrt{b^{3}} + b^{2}$
No. Because, there is a square root over $b$ . So, there is fractional power over a variable.

(c) $\frac{5}{2} x^{2} - 4 y z - \frac{5}{6} x^{2}$
Yes. Because although the coefficient $\frac{5}{2}$ and $- \frac{5}{6}$ are fractional coefficients, the powers of variables are all whole numbers.

(d) $3 p^{3} - 4 q^{\frac{2}{3}} + p q r + 3 p q$
No. Because there is a fractional power over $q$ .

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