Are the following expressions polynomials?

(1) 21 – 8a + a²

(2) n³ – 6n² + 13

(3)

(4)

(5)

(6)

(7) 51

(8) 9 – 5b² + b⁴

(9)

(1) The given algebraic expression is a polynomial in one variable because all the powers of x are whole numbers.

In this given polynomial the power of a is +2, 1 and 0.

Hence these are whole powers of a and hence this expression is a polynomial.

(2) The given algebraic expression is a polynomial in one variable because all the powers of x are whole numbers.

In this given polynomial the power of n is +3, 2 and 0.

Hence these are whole powers of n and hence this expression is a polynomial.

(3) The given algebraic expression is not a polynomial because all the powers of x are not whole numbers.

In this given polynomial the power of x is 2 and 3.5.

Hence these 3.5 is not a whole power of x and hence this expression is not a polynomial.

(4) The number -13/9 is a polynomial because 10 can be represented as follows: = 13/9 × x⁰ which is equal to -13/9. So it is a polynomial.

(5) n - 1/n is not a polynomial because in this given equation the second term becomes 1 × n^-1 where -1 is not a whole number. So the given algebraic expression is not a polynomial.

(6) The given algebraic expression is a polynomial in one variable because all the powers of x are whole numbers.

In this given polynomial the power of a is +3, 1 and 0.

Hence these are whole powers of a and hence this expression is a polynomial.

(7) The number 51 is a polynomial because 10 can be represented as follows: 51 × x⁰ which is equal to 51. So it is a polynomial.

(8) The given algebraic expression is a polynomial in one variable because all the powers of x are whole numbers.

In this given polynomial the power of b is 4, 2 and 0.

Hence these are whole powers of b and hence this expression is a polynomial.

(9) This expression can be expressed as follows:

12 + 10p²

The given algebraic expression is a polynomial in one variable because all the powers of x are whole numbers.

In this given polynomial the power of p is 2 and 0.

Hence these are whole powers of p and hence this expression is a polynomial.