(1)  Solution :

State which of the following expressions are polynomials in one variable or not. Give reasons for your answer.

(v)  ∛t+2t

Since the exponent of t is not a whole number, it is not a polynomial.

(vi)  x³ + y³ +z⁶

Not a polynomial in one variable.

(2)  Solution :

(3)  Solution :

(4)  Solution :

(i)  3x² + 2x +1

Since the degree of the polynomial is 2, it is a quadratic equation(polynomial).

(ii)  4x³-1

Since the degree of the polynomial is 3, it is a cubic equation(polynomial).

(iii)  y+3

Since the degree of the polynomial is 1, it is a linear polynomial.

(iv)  y² - 4

Since the degree of the polynomial is 2, it is a quadratic polynomial.

(v)  4x³

Since the degree of the polynomial is 3, it is a cubic polynomial.

(vi) 2x

Since the degree of the polynomial is 1, it is a linear polynomial.

(5)  Solution :

Give one example of a binomial of degree 27 and monomial of degree 49 and trinomial of degree 36.

• Binomial of degree 27  =  ax²⁷+b
• Monomial of degree  49  =  cy⁴⁹
• Trinomial of degree 36  =  ax³⁶+ bx⁶ + cx

(6) Solution :

a) 4w²

The highest exponent of the polynomial is 2,

Degree = 2

Number of terms = 1

b) 9x³ + 2x

The highest exponent of the polynomial is 3,

Degree = 3

Number of terms = 2

c) 4p⁵ - p³ + p² + 11

The highest exponent of the polynomial is 2,

Degree = 5

Number of terms = 4

(7) Solution :

(a)  The polynomial which has three terms is called trinomial. So,

A trinomial is a polynomial with three terms

b) The polynomial which has one term is called monomial. So,

A monomial is a polynomial with one term.

c) A polynomial which has two terms is called binomial. So,

A binomial is polynomial with two terms.

(8) Writing the polynomial which has the highest exponent to the lowest exponent is known as writing down the polynomial in standard form.

a) 12 + 3x² - x

Standard form :

3x² - x + 12

• The highest exponent is 2, so degree is 2.
• Coefficient of highest exponent term is 3, so leading coefficient is 3.

b) g⁴ - 2g³ - g⁵

Standard form :

- g⁵ + g⁴ - 2g³

• The highest exponent is 5, so degree is 5.
• Coefficient of highest exponent term is -1, so leading coefficient is -1.

c) k² + k⁴ - k⁵ + 1

Standard form :

- k⁵ - k⁴ + k²  + 1

• The highest exponent is 5, so degree is 5.
• Coefficient of highest exponent term is -1, so leading coefficient is -1.

(9)  Solution :

a) Monomial will have only one term, in -3xy⁴

Degree of -3xy⁴ = 1 + 4 ==> 5

b) Since the given is sum of monomials, we consider the sum of two monomials 

2x⁴ y⁴ + 3x²y⁵

degree = 4 + 4 ==> 8

Since the highest exponent can be considered as degree, the degree of the above polynomial is 8.

c)  Trinomial will consist of 3 terms,

x⁵ + x³ + 1

The degree of the polynomial is 5.

d)  2xy + 5xz 

e)  2x⁴

f)  4xyz² + xyz

g) In the polynomial 4x² - 3x⁵ + x

Writing in standard form, we get

- 3x⁵ +  4x² + x

The leading coefficient is -3

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