Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (i) 4x^{2} – 3x + 7 (ii) y^{2} + √2 (iii) 3 √t + t√2 (iv) y+ \(\frac { 2 }{ y }\) (v) x^{10}+ y^{3}+t^{50}

Solution:

(i) We have 4x^{2} – 3x + 7 = 4x^{2} – 3x + 7x^{0}.
It is a polynomial in one variable i.e. x
Because every exponent of x is a whole number.

(ii) We have y^{2} + √2 = y^{2} + √2y^{0} .
It is a polynomial in one variable, i.e., y
Because every exponent of y is a whole number.

(iii) We have 3 √t + t√2 = 3 √t^{1/2} + √2.t
It is not a polynomial, because one of the exponents of t is \(\frac { 1 }{ 2 }\),
which is not a whole number.

(iv) We have y + \(y+\frac { 2 }{ y }\) = y + 2.y^{-1}
It is not a polynomial, because one of the exponents of y is -1,
which is not a whole number.

(v) We have x^{10}+ y^{3 }+ t^{50}
Here, the exponent of each variable is a whole number, but x^{10}+ y^{3 }+ t^{50} is a polynomial in x, y and t, that is, in three variables.
Hence it is not a polynomial with one variable.

Ex 2.1 Class 9 Maths Question 2.

Write the coefficients of x^{2} in each of the following (i) 2 + x^{2} + x (ii) 2 – x^{2} + x^{3} (iii) \(\frac { \pi }{ 2 }\) x^{2} + x (iv) √2 x – 1

Solution:

(i) The given polynomial is 2 + x^{2} + x.
The coefficient of x^{2} is 1.
(ii) The given polynomial is 2 – x^{2} + x^{3}.
The coefficient of x^{2} is -1.
(iii) The given polynomial is \(\frac { \pi }{ 2 } { x }^{ 2 }\) + x.
The coefficient of x^{2} is \(\frac { \pi }{ 2 }\) .
(iv) The given polynomial is √2 x – 1.
The coefficient of x^{2} is 0.

Ex 2.1 Class 9 Maths Question 3.

Give one example each of a binomial of degree 35, and of a monomial of degree 100. Solution:

(i) Abmomial of degree 35 can be 3x^{35} -4.
(ii) A monomial of degree 100 can be √2y^{100}.

Ex 2.1 Class 9 Maths Question 4.

Write the degree of each of the following polynomials. (i) 5x^{3}+4x^{2} + 7x (ii) 4 – y^{2} (iii) 5t – √7 (iv) 3 Solution:

(i) The given polynomial is 5x^{3} + 4x^{2} + 7x
The highest power of the variable x is 3.
Hence, the power of the polynomial is 3.
(ii) The given polynomial is 4- y^{2}. Highest
The power of the variable y is 2.
Hence, the power of the polynomial is 2.
(iii) The given polynomial is 5t – √7 . The highest power of the variable t is 1. Hence, the power of the polynomial is 1.
(iv) Since,3 = 3x° [∵ x°=1]
So the power of the polynomial is 0.

Ex 2.1 Class 9 Maths Question 5.

Classify the following as linear, quadratic and cubic polynomials. (i) x^{2}+ x (ii) x – x^{3} (iii) y + y^{2}+4 (iv) 1 + x (v) 3t (vi) r^{2} (vii) 7x^{3}

^{
}Solution:

(i) The power of x^{2} + x is 2. Hence, it is a quadratic polynomial.
(ii) The power of x – x^{3} is 3. Hence, it is a cubic polynomial.

(iii) The power of y + y^{2} + 4 is 2. Hence, it is a quadratic polynomial.

(iv) The power of 1 + x is 1. Hence it is a linear polynomial.
(v) The power of 3t is 1. Hence, it is a linear polynomial.
(vi) The power of r^{2} is 2. Hence, it is a quadratic polynomial.
(vii) The power of 7x^{3} is 3. Hence, it is a cubic polynomial.