Let us write which are the polynomials in the following algebraic expressions. Let us write the degree of each of the polynomials.

(i) 2x⁶ – 4x⁵ + 7x² + 3

(ii) x^–2 + 2x^–1 + 4

(iii)

(iv)

(v) x⁵¹ – 1

(vi)

(vii) 15

(viii) 0

(ix)

(x) y³ + 4

(xi)

(i) Since all the exponents (power) of variable(x) are whole no. (i.e., zero or positive integers)

⇒ 2x⁶ – 4x⁵ + 7x² + 3 is a polynomial.

(Note: The degree is the value of the greatest (highest) exponent of any expression (except the constant) in the polynomial. To find the degree all that you have to do is find the largest exponent in the polynomial).

And, since here the highest power is 6,

⇒ It is a polynomial of degree 6.

(ii) Since all the exponents (power) of variable(x) are not whole no. (i.e., zero or positive integers)

⇒ x^–2 + 2x^–1 + 4 is not a polynomial.

(iii) Since all the exponents (power) of variable(y) are whole no. (i.e., zero or positive integers)

⇒ is a polynomial.

And, since here the highest power is 3,

⇒ It is a polynomial of degree 3.

(iv)

Since all the exponents (power) of variable(x) are not whole no.(i.e., zero or positive integers)

is not a polynomial.

(v) Since all the exponents (power) of variable(x) are whole no. (i.e., zero or positive integers)

⇒ x⁵¹ – 1 is a polynomial.

And, since here the highest power is 51,

⇒ x⁵¹ – 1 is a polynomial of degree 51.

(vi)

Since all the exponents (power) of t are not whole no.(i.e., zero or positive integers)

is not a polynomial.

(vii) 15 is a constant polynomial as there is no variable present in that.

And, therefore the highest power is 0

⇒ 15 is a polynomial with degree 0.

(viii) 0 is a constant polynomial (zero polynomial) as there is no variable present in that.

⇒ 0 is a polynomial with degree 0.

(ix)

Since all the exponents (power) of variable(z) are not whole no.(i.e., zero or positive integers)

is not a polynomial.

(x) Since all the exponents (power) of variable(y) are whole no. (i.e., zero or positive integers)

⇒ y³ +4 is a polynomial.

And, since here the highest power is 3,

⇒ y³ +4 is a polynomial of degree 3.

(xi)

Since all the exponents (power) of variable(x) are whole no.(i.e., zero or positive integers)

And, since here the highest power is 2

is a polynomial of degree 2.