Exercise 2.1

1.  Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x² = 3x + 7

Ans.- $4{x^2} - 3x + 7$ is a polynomial in variable x, as exponents of the variable in different terms are whole numbers.

(ii) y² + $\sqrt 2$

Ans.- ${y^2} + \sqrt 2$ is a polynomial in variable y, as exponent of variable y is a whole number.

(iii) $3\sqrt t + t\sqrt 2$

Ans.- $3\sqrt t + t\sqrt 2 = 3{t^{{1 \over 2}}} + \sqrt 2$ t is not a polynomial, as exponent of variable t in first term, i.e., 3 t 1/2 is not a whole number.

(iv) $y + {2 \over y}$

Ans.- $y + {2 \over y} = y + 2{y^{ - 1}}$ is not a polynomial, as exponent of the variable y in the term 2y-1  is not a whole number.

(v) ${x^{10}} + {y^3} + {t^{50}}$

Ans.-  x10 + y3 + t50 is a polynomial in three variables.

2. Write the coefficient of x2 in each of the following :
(i) 2 + x2 + x

Ans. –   Coefficient of x² in 2 + x² + x is 1 .

(ii) 2 – x2 + x3

Ans. –  Coefficient of x² in 2 – x² + x³ is – 1.

(iii) ${\pi \over 2}{x^2} + x$

Ans. –  Coefficient of x² in ${\pi \over 2}{x^2} + x$  is ${\pi \over 2}$

(iv)  $\sqrt 2 x - 1$

Ans. – Coefficient of x² in $\sqrt 2$ – 1 is 0.

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

Sol.
(i) Binomial of degree 35 is 2×35 + x.
(ii) Monomial of degree 100 is – 7×100

4. Write the degree of each of the following polynomials:
(i) 5×3 + 4×2 + 7x

Ans. – Degree of polynomial  $5{x^3} + 4{x^2} + 7x$ is 3.

(ii) 4 – y2

Ans. – Degree of polynomial 4 – y2 is 2.

(iii) 5t – $\sqrt 7$

Ans. – Degree of polynomial 5t – $\sqrt 7$  is 1.

(iv)

Ans. – Degree of polynomial 3 is o.

5. Classify the following as linear, quadratic and cubic polynomials :
(i) x2 + x

Ans. –  Polynomial x2 +x ia a quadratic polynomial.

(ii) x-x3

Ans. – Polynomial x – x³ is a cubic polynomial.

(iii) y + y2+4

Ans. – Polynomial  y + y² + 4  is a quadratic polynomial.

(iv)  1 + x

Ans. – Polynomial 1 + x is a linear polynomial.

(v) 3t

Ans. – Polynomial 3t is a linear polynomial.

(vi) r2

Ans. – Polynomial r2 is a quadratic polynomial.

(vii) 7×3