Polynomials
NCERT Solutions • Class 9 Maths • Chapter 2Exercise 2.1
1. Which of the following expressions are polynomials in one variable and which are not? State reasons.
(i) $4x^2 – 3x + 7$
Polynomial Reason: All exponents of $x$ (2, 1, 0) are whole numbers.
Polynomial Reason: All exponents of $x$ (2, 1, 0) are whole numbers.
(ii) $y^2 + \sqrt{2}$
Polynomial Reason: Exponent of $y$ is 2 (whole number). Constant $\sqrt{2}$ is allowed.
Polynomial Reason: Exponent of $y$ is 2 (whole number). Constant $\sqrt{2}$ is allowed.
(iii) $3\sqrt{t} + t\sqrt{2}$
Not a Polynomial Reason: $3\sqrt{t} = 3t^{1/2}$. The power $\frac{1}{2}$ is not a whole number.
Not a Polynomial Reason: $3\sqrt{t} = 3t^{1/2}$. The power $\frac{1}{2}$ is not a whole number.
(iv) $y + \frac{2}{y}$
Not a Polynomial Reason: $\frac{2}{y} = 2y^{-1}$. The power $-1$ is negative, not a whole number.
Not a Polynomial Reason: $\frac{2}{y} = 2y^{-1}$. The power $-1$ is negative, not a whole number.
(v) $x^{10} + y^3 + t^{50}$
Not in one variable Reason: It has 3 variables ($x, y, t$), so it’s not in one variable.
Not in one variable Reason: It has 3 variables ($x, y, t$), so it’s not in one variable.
2. Write the coefficients of $x^2$ in each of the following:
(i) $2 + x^2 + x$
Term containing $x^2$: $1 \cdot x^2$ $\rightarrow$ Coefficient: 1
Term containing $x^2$: $1 \cdot x^2$ $\rightarrow$ Coefficient: 1
(ii) $2 – x^2 + x^3$
Term containing $x^2$: $-1 \cdot x^2$ $\rightarrow$ Coefficient: -1
Term containing $x^2$: $-1 \cdot x^2$ $\rightarrow$ Coefficient: -1
(iii) $\frac{\pi}{2}x^2 + x$
Term containing $x^2$: $\frac{\pi}{2} \cdot x^2$ $\rightarrow$ Coefficient: $\frac{\pi}{2}$
Term containing $x^2$: $\frac{\pi}{2} \cdot x^2$ $\rightarrow$ Coefficient: $\frac{\pi}{2}$
(iv) $\sqrt{2}x – 1$
No $x^2$ term (can be written as $0 \cdot x^2$) $\rightarrow$ Coefficient: 0
No $x^2$ term (can be written as $0 \cdot x^2$) $\rightarrow$ Coefficient: 0
3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Binomial (Degree 35)
Example: $x^{35} + 10$ (Polynomial with 2 terms, highest power 35)
Example: $x^{35} + 10$ (Polynomial with 2 terms, highest power 35)
Monomial (Degree 100)
Example: $\sqrt{2}y^{100}$ (Polynomial with 1 term, highest power 100)
Example: $\sqrt{2}y^{100}$ (Polynomial with 1 term, highest power 100)
4. Write the degree of each of the following polynomials:
(i) $5x^3 + 4x^2 + 7x$
Highest Power: 3 $\rightarrow$ Degree: 3
Highest Power: 3 $\rightarrow$ Degree: 3
(ii) $4 – y^2$
Highest Power: 2 $\rightarrow$ Degree: 2
Highest Power: 2 $\rightarrow$ Degree: 2
(iii) $5t – \sqrt{7}$
Highest Power: 1 ($t^1$) $\rightarrow$ Degree: 1
Highest Power: 1 ($t^1$) $\rightarrow$ Degree: 1
(iv) $3$
Highest Power: 0 ($3x^0$) $\rightarrow$ Degree: 0
Highest Power: 0 ($3x^0$) $\rightarrow$ Degree: 0
5. Classify the following as linear, quadratic and cubic polynomials:
(i) $x^2 + x$
Degree 2 $\rightarrow$ Quadratic
Degree 2 $\rightarrow$ Quadratic
(ii) $x – x^3$
Degree 3 $\rightarrow$ Cubic
Degree 3 $\rightarrow$ Cubic
(iii) $y + y^2 + 4$
Degree 2 $\rightarrow$ Quadratic
Degree 2 $\rightarrow$ Quadratic
(v) $3t$
Degree 1 $\rightarrow$ Linear
Degree 1 $\rightarrow$ Linear
(vi) $r^2$
Degree 2 $\rightarrow$ Quadratic
Degree 2 $\rightarrow$ Quadratic
(vii) $7x^3$
Degree 3 $\rightarrow$ Cubic
Degree 3 $\rightarrow$ Cubic