Question

Are the following expressions polynomials?

(1) 21 − 8a + a²
(2) n³ − 6n² + 13
(3) $8x^2-x^{\frac{7}{2}}$
(4) $\frac{-13}{9}$
(5) $n-\frac{1}{n}$
(6) $\frac{2}{5}{a}^3-7a+19$
(7) 51
(8) 9 − 5b² + b⁴
(9) $12+\frac{10}{p^{-2}}$

Answer 1

(1) The powers 0, 1 and 2 of the variable a in 21 − 8a + a² are whole numbers.
Hence, 21 − 8a + a² is a polynomial.

(2) The powers 3, 2 and 0 of the variable n in n³ − 6n² + 13 are whole numbers.
Hence, n³ − 6n² + 13 is a polynomial.

(3) In the expression $8x^2-x^{\frac{7}{2}}$, the index $\frac{7}{2}$ is not a whole number.
Hence, this expression is not a polynomial.

(4) The power 0 of the variable x in $\frac{-13}{9}\left(i.e.\frac{-13}{9}{x}^0\right)$ is a whole number.
Hence $\frac{-13}{9}$ is a polynomial.
Note: Every constant is a polynomial and it can be written in terms of any variable with power zero.

(5) In the expression $n-\frac{1}{n},\left(i.e.n-n^{-1}\right)$, the index −1 is not a whole number.
Hence, this expression is not a polynomial.

(6) The powers 3, 1 and 0 of the variable a in $\frac{2}{5}{a}^3-7a+19$ are whole numbers.
Hence, $\frac{2}{5}{a}^3-7a+19$ is a polynomial.

(7) The power 0 of the variable x in 51 (i.e. 51 x⁰) is a whole number.
Hence, 51 is a polynomial.
Note: Every constant is a polynomial and it can be written in terms of any variable with power zero.

(8) The powers 0, 2 and 4 of the variable b in 9 − 5b² + b⁴ are whole numbers.
Hence, 9 − 5b² + b⁴ is a polynomial.

(9) The powers 0 and 2 of the variable p in $12+\frac{10}{p^{-2}}\left(i.e.12+10p^2\right)$ are whole numbers.
Hence, $12+\frac{10}{p^{-2}}$ is a polynomial.