Home > NCERT Solutions > NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials

NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials

Polynomials is a core and foundational chapter in Class 9 Mathematics that introduces algebraic expressions used across higher classes. This chapter covers definition of polynomials, terms, coefficients, degree of a polynomial, types of polynomials, value of a polynomial, and zeros of a polynomial, which are essential for building strong algebra skills.

NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials are prepared strictly according to the latest CBSE syllabus and exam pattern. The solutions are explained in simple, step-by-step language with clear methods and worked examples, helping students understand concepts easily and score well in Class 9 exams.

NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials

Polynomials

Medium

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

\begin{array}{l} (i) {4x}^{2} - 3x+7 \\ (ii) y^{2} + \sqrt{2} \\ (iii) 3 \sqrt{t} +t \sqrt{2} \\ (iv) y+ \frac{2}{y} (v) x^{10} {+y}^{3} {+t}^{50} \end{array}

Polynomials

Medium

\begin{array}{l} Factorise​ the following using appropriate identities: \\ (i) {9x}^{2} {+6xy+y}^{2} \\ (ii) {4y}^{2} - 4y+1 \\ (iii) x^{2} - \frac{y^{2}}{100} \end{array}

Polynomials

Difficult

\begin{array}{l} Give possible expressions for the length and breadth of \\ each of the following rectangles, in which their areas are \\ given: \\ (i) {Area : 25a}^{2} -35a+12 \\ (ii) {Area : 35y}^{2} +13y-12 \end{array}

Polynomials

Medium

Without actually calculating the cubes, find the value of each of the following:
(i) (–12)³ + (7)³ + (5)³
(ii) (28)³ + (–15)³ + (–13)³

Polynomials

Difficult

\begin{array}{l} Verify that : \\ x^{3} + y^{3} + z^{3} - 3 xyz = \frac{1}{2} (x + y + z) [{(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2}] \end{array}

Polynomials

Difficult

Factorise each of the following:
(i) 27y³+ 125z³
(ii) 64m³ – 343n³

Polynomials

Medium

\begin{array}{l} Write​ the following cubes in expanded form: \\ (i) {(2x+1)}^{3} \\ (ii) {(2a-3b)}^{3} \\ (iii) {[\frac{3}{2} x+1]}^{3} \\ (iv) {[x- \frac{2}{3} y]}^{3} \end{array}

Polynomials

Medium

\begin{array}{l} Factorise : \\ (i) 4 x^{2} + 9 y^{2} + 16 z^{2} + 12 xy - 24 yz - 16 xz \\ (ii) 2 x^{2} + y^{2} + 8 z^{2} - 2 \sqrt{2} xy + 4 \sqrt{2} yz - 8 xz \end{array}

Polynomials

Difficult

\begin{array}{l} Expand each of the following, using suitable identities: \\ (i) {(x+2y+4z)}^{2} \\ (ii) {(2x - y+z)}^{2} \\ (iii) {(- 2x+3y+2z)}^{2} \\ (iv) {(3a - 7b - c)}^{2} \\ (vi) {(- 2x+5y - 3z)}^{2} \\ (vii) {[\frac{1}{4} a - \frac{1}{2} b+1]}^{2} \end{array}

Polynomials

Difficult

\begin{array}{l} Use suitable identities to find the following products: \\ (i) (x+4) (x+10) \\ (ii) (x+8) (x - 10) \\ (iii) (3x+4) (3x - 5) \\ (iv) (y^{2} + \frac{3}{2}) (y^{2} - \frac{3}{2}) \\ (v) (3 - 2x) (3+2x) \end{array}

Polynomials

Medium

\begin{array}{l} {Write the coefficients of x}^{2} in each of the following: \\ (i) {2+x}^{2} +x \\ (ii) 2 - x^{2} {+x}^{3} \\ (iii) \frac{π}{2} x^{2} +x \\ (iv) \sqrt{2} x - 1 \end{array}

Polynomials

Difficult

\begin{array}{l} Find the value of k, if (x-1) is a factor of p (x) in each of the \\ following cases: \\ (i) p (x) {= x}^{2} +x+k \\ (ii) p (x) {= 2x}^{2} +kx+ \sqrt{2} \\ (iii) p (x) {= kx}^{2} - \sqrt{2} x+1 \\ (iv) p (x) {= kx}^{2} - 3x+k \end{array}

Polynomials

Difficult

\begin{array}{l} Determine which of the following polynomials has (x+1) a \\ factor: \\ (i) x^{3} {+x}^{2} +x+1 \\ (ii) x^{4} {+x}^{3} {+x}^{2} +x+1 \\ (iii) x^{4} {+3x}^{3} {+3x}^{2} +x+1 \\ (iv) x^{3} - x^{2} - (2+ \sqrt{2}) x+ \sqrt{2} \end{array}

Polynomials

Difficult

\begin{array}{l} {Find the remainder when x}^{3} {+3x}^{2} +3x+1 is divided by \\ (i) x+1 \\ (ii) x - \frac{1}{2} \\ (iii) x \\ (iv) x+π \\ (v) 5+2x \end{array}

Polynomials

Difficult

\begin{array}{l} Verify whether the following are zeroes of the \\ polynomial, indicated against them. \\ (i) p (x) = 3 x + 1, x = - \frac{1}{3} \\ (ii) p (x) = 5 x - π, x = \frac{4}{5} \\ (iii) p (x) = x^{2} - 1, x = 1, - 1 \\ (iv) p (x) = (x + 1) (x - 2), x = - 1, 2 \\ (v) p (x) = x^{2}, x = 0 \\ (vi) p (x) = lx + m, x = - \frac{m}{l} \\ (vii) p (x) = 3 x^{2} - 1, x = - \frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}} \\ (viii) p (x) = 2 x + 1, x = \frac{1}{2} \end{array}

Polynomials

Difficult

Find p(0), p(1) and p(2) for each of the following polynomials:

(i) p(y) = y² – y + 1 (ii) p(t) = 2 + t + 2t² – t³

(iii) p(x) = x³ (iv) p(x) = (x – 1) (x + 1)

Polynomials

Medium

Find the value of the polynomial 5x – 4x² + 3 at
(i) x = 0
(ii) x = –1
(iii) x = 2

Polynomials

Medium

\begin{array}{l} 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) 3 t (vi) r^{2} (vii) 7 x^{3} \end{array}

Polynomials

Medium

\begin{array}{l} Write the degree of each of the following polynomials: \\ (i) {5x}^{3} {+4x}^{2} +7x \\ (ii) 4 - y^{2} \\ (iii) 5t - \sqrt{7} \\ (iv) 3 \end{array}

Polynomials

Difficult

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?
(i) Volume : 3x²–12x
(ii) Volume: 12ky² + 8ky – 20k

NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials

1) Which of the following expressions are polynomials in one variable and which are not? Give reasons.

Answer:

$4x^2 - 3x + 7$

$4 x^{2} - 3 x + 7$
✅ Polynomial in one variable (x)
Reason: All powers of x are non-negative integers.

ii)

$y^2 + 2$

$y^{2} + 2$
✅ Polynomial in one variable (y)

iii)

$3t + t^2$

$3 t + t^{2}$
✅ Polynomial in one variable (t)

iv)

$y + \frac{2}{y}$

$y + y 2$
❌ Not a polynomial
Reason: Variable appears in the denominator (negative power).

$x^{10} + y^3 + t^{50}$

$x^{10} + y^{3} + t^{50}$
❌ Not a polynomial in one variable
Reason: More than one variable is involved.

2) Write the coefficient of $x^2$

$x^{2}$ in each of the following:

Answer:

$2 + x^2 + x$

$2 + x^{2} + x$ → 1
ii)

$2 - x^2 + x^3$

$2 - x^{2} + x^{3}$ → –1
iii)

$\frac{\pi}{2}x^2 + x$

$2 π x^{2} + x$ →

$\frac{\pi}{2}$

$2 π$
iv)

$2x - 1$

$2 x - 1$ → 0 (no

$x^2$

$x^{2}$ term)

3) Write the degree of each polynomial:

Answer:

$5x^3 + 4x^2 + 7x$

$5 x^{3} + 4 x^{2} + 7 x$ → 3
ii)

$4 - y^2$

$4 - y^{2}$ → 2
iii)

$5t - 7$

$5 t - 7$ → 1
iv)

$3$

$3$ → 0 (constant polynomial)

4) Classify the following polynomials as linear, quadratic or cubic:

Answer:

$x^2 + x$

$x^{2} + x$ → Quadratic
ii)

$x - x^3$

$x - x^{3}$ → Cubic
iii)

$y + y^2 + 4$

$y + y^{2} + 4$ → Quadratic
iv)

$1 + x$

$1 + x$ → Linear
v)

$3t$

$3 t$ → Linear
vi)

$r^2$

$r^{2}$ → Quadratic
vii)

$7x^3$

$7 x^{3}$ → Cubic

5) Find the value of the polynomial $5x - 4x^2 + 3$

$5 x - 4 x^{2} + 3$ :

Answer:

i) At

$x = 0$

$x = 0$ : 3
ii) At

$x = -1$

$x = - 1$ : –6
iii) At

$x = 2$

$x = 2$ : –3

6) Find p(0), p(1) and p(2):

Answer:

$p(y) = y^2 - y + 1$

$p (y) = y^{2} - y + 1$
p(0) = 1, p(1) = 1, p(2) = 3

ii)

$p(t) = 2 + t + 2t^2 - t^3$

$p (t) = 2 + t + 2 t^{2} - t^{3}$
p(0) = 2, p(1) = 4, p(2) = 4

iii)

$p(x) = x^3$

$p (x) = x^{3}$
p(0) = 0, p(1) = 1, p(2) = 8

iv)

$p(x) = (x - 1)(x + 1)$

$p (x) = (x - 1) (x + 1)$
p(0) = –1, p(1) = 0, p(2) = 3

7) Verify whether the given values are zeroes of the polynomial:

Answer:

$3x + 1$

$3 x + 1$ , at

$x = -\frac13$

$x = - 3 1$ → ✔ Zero
ii)

$5x - \pi$

$5 x - π$ , at

$x = \frac45$

$x = 5 4$ → ✘ Not a zero
iii)

$x^2 - 1$

$x^{2} - 1$ , at

$x = 1, -1$

$x = 1, - 1$ → ✔✔
iv)

$(x + 1)(x - 2)$

$(x + 1) (x - 2)$ , at

$x = -1, 2$

$x = - 1, 2$ → ✔✔
v)

$x^2$

$x^{2}$ , at

$x = 0$

$x = 0$ → ✔
vi)

$lx + m$

$l x + m$ , at

$x = -\frac{m}{l}$

$x = - l m$ → ✔
vii)

$3x^2 - 1$

$3 x^{2} - 1$ , at

$x = -\frac13, \frac23$

$x = - 3 1, 3 2$ → ✘✘
viii)

$2x + 1$

$2 x + 1$ , at

$x = \frac12$

$x = 2 1$ → ✘

8) Find the remainder when $x^3 + 3x^2 + 3x + 1$

$x^{3} + 3 x^{2} + 3 x + 1$ is divided by:

👉 Note:

$x^3 + 3x^2 + 3x + 1 = (x + 1)^3$

$x^{3} + 3 x^{2} + 3 x + 1 = (x + 1)^{3}$

Answer:

i) By

$x + 1$

$x + 1$ : 0
ii) By

$x - \frac12$

$x - 2 1$ :

$\frac{27}{8}$

$8 27$
iii) By

$x$

$x$ : 1
iv) By

$x + \pi$

$x + π$ :

$(1 - \pi)^3$

$(1 - π)^{3}$
v) By

$5 + 2x$

$5 + 2 x$ : –

$\frac{27}{8}$

$8 27$

9) Determine which polynomial has $x + 1$

$x + 1$ as a factor:

Answer:

$x^3 + x^2 + x + 1$

$x^{3} + x^{2} + x + 1$ → ✔
ii)

$x^4 + x^3 + x^2 + x + 1$

$x^{4} + x^{3} + x^{2} + x + 1$ → ✘
iii)

$x^4 + 3x^3 + 3x^2 + x + 1$

$x^{4} + 3 x^{3} + 3 x^{2} + x + 1$ → ✘
iv) Given expression → ✘

10) Find the value of k if $x - 1$

$x - 1$ is a factor:

Answer:

$x^2 + x + k$

$x^{2} + x + k$ → k = –2
ii)

$2x^2 + kx + 2$

$2 x^{2} + k x + 2$ → k = –4
iii)

$kx^2 - 2x + 1$

$k x^{2} - 2 x + 1$ → k = 1
iv)

$kx^2 - 3x + k$

$k x^{2} - 3 x + k$ → k =

$\frac32$

$2 3$

11) Use identities to find the products:

Answer:

$(x + 4)(x + 10) = x^2 + 14x + 40$

$(x + 4) (x + 10) = x^{2} + 14 x + 40$
ii)

$(x + 8)(x - 10) = x^2 - 2x - 80$

$(x + 8) (x - 10) = x^{2} - 2 x - 80$
iii)

$(3x + 4)(3x - 5) = 9x^2 - 3x - 20$

$(3 x + 4) (3 x - 5) = 9 x^{2} - 3 x - 20$
iv)

$(y^2 + 3)(y^2 - 3) = y^4 - 9$

$(y^{2} + 3) (y^{2} - 3) = y^{4} - 9$
v)

$(3 - 2x)(3 + 2x) = 9 - 4x^2$

$(3 - 2 x) (3 + 2 x) = 9 - 4 x^{2}$

12) Factorise using identities:

Answer:

$9x^2 + 6xy + y^2 = (3x + y)^2$

$9 x^{2} + 6 x y + y^{2} = (3 x + y)^{2}$
ii)

$4y^2 - 4y + 1 = (2y - 1)^2$

$4 y^{2} - 4 y + 1 = (2 y - 1)^{2}$
iii)

$x^2 - \frac{y^2}{100} = \left(x - \frac{y}{10}\right)\left(x + \frac{y}{10}\right)$

$x^{2} - 100 y ^{2} = (x - 10 y) (x + 10 y)$

13) Expand using identities:

(Standard expansion using

$(a+b+c)^2$

$(a + b + c)^{2}$ and

$(a-b-c)^2$

$(a - b - c)^{2}$ )

14) Factorise:

Answer:

$4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16xz = (2x + 3y - 4z)^2$

$4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 yz - 16 x z = (2 x + 3 y - 4 z)^{2}$

ii)

$2x^2 + y^2 + 8z^2 - 2\sqrt2xy + 4\sqrt2yz - 8xz = (\sqrt2x - y + 2\sqrt2z)^2$

$2 x^{2} + y^{2} + 8 z^{2} - 22 x y + 42 yz - 8 x z = (2 x - y + 22 z)^{2}$

15) Write the cubes in expanded form:

Answer:

$(2x + 1)^3$

$(2 x + 1)^{3}$
ii)

$(2a - 3b)^3$

$(2 a - 3 b)^{3}$
iii)

$\left(\frac32x + 1\right)^3$

$(2 3 x + 1)^{3}$
iv)

$\left(x - \frac23y\right)^3$

$(x - 3 2 y)^{3}$

16) Factorise:

Answer:

$27y^3 + 125z^3 = (3y + 5z)(9y^2 - 15yz + 25z^2)$

$27 y^{3} + 125 z^{3} = (3 y + 5 z) (9 y^{2} - 15 yz + 25 z^{2})$
ii)

$64m^3 - 343n^3 = (4m - 7n)(16m^2 + 28mn + 49n^2)$

$64 m^{3} - 343 n^{3} = (4 m - 7 n) (16 m^{2} + 28 mn + 49 n^{2})$

17) Verify the identity:

$x^3 + y^3 + z^3 - 3xyz = \frac12 (x + y + z)\big[(x - y)^2 + (y - z)^2 + (z - x)^2\big]$

$x^{3} + y^{3} + z^{3} - 3 x yz = 21 (x + y + z) [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]$

Answer:

✔ LHS = RHS, identity verified.

18) Find the value without calculating cubes:

Answer:

$(-12)^3 + 7^3 + 5^3 = 0$

$(- 12)^{3} + 7^{3} + 5^{3} = 0$
ii)

$28^3 + (-15)^3 + (-13)^3 = 0$

$2 8^{3} + (- 15)^{3} + (- 13)^{3} = 0$

19) Find possible expressions for dimensions of rectangles:

Answer:

i) Area =

$25a^2 - 35a + 12 = (5a - 3)(5a - 4)$

$25 a^{2} - 35 a + 12 = (5 a - 3) (5 a - 4)$
ii) Area =

$35y^2 + 13y - 12 = (7y - 3)(5y + 4)$

$35 y^{2} + 13 y - 12 = (7 y - 3) (5 y + 4)$

20) Find possible expressions for dimensions of cuboids:

Answer:

i) Volume =

$3x^2 - 12x = 3x(x - 4)$

$3 x^{2} - 12 x = 3 x (x - 4)$
ii) Volume =

$12ky^2 + 8ky - 20k = 4k(3y - 2)(y + 1)$

$12 k y^{2} + 8 k y - 20 k = 4 k (3 y - 2) (y + 1)$

FAQs: Class 9 Maths Chapter 2 – Polynomials

Q1. Is Polynomials important for Class 9 exams?
Yes, it is a foundational algebra chapter with regular exam questions.

Q2. Which topics are most important in this chapter?
Degree of polynomials, types of polynomials, and zeros of a polynomial.

Q3. Are graphical questions asked from this chapter?
Yes, questions on zeros of a polynomial using graphs are common.

Q4. Is this chapter important for Class 10 Maths?
Yes, it forms the base for Class 10 Polynomials and Quadratic Equations.

Q5. How do NCERT Solutions help?
They provide NCERT-aligned, exam-ready explanations with solved examples.

NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials

NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials

NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials

1) Which of the following expressions are polynomials in one variable and which are not? Give reasons.

Answer:

2) Write the coefficient of x2x^2

x2 in each of the following:

Answer:

3) Write the degree of each polynomial:

Answer:

4) Classify the following polynomials as linear, quadratic or cubic:

Answer:

5) Find the value of the polynomial 5x−4x2+35x - 4x^2 + 3

5x−4x2+3:

Answer:

6) Find p(0), p(1) and p(2):

Answer:

7) Verify whether the given values are zeroes of the polynomial:

Answer:

8) Find the remainder when x3+3x2+3x+1x^3 + 3x^2 + 3x + 1

x3+3x2+3x+1 is divided by:

Answer:

9) Determine which polynomial has x+1x + 1

x+1 as a factor:

Answer:

10) Find the value of k if x−1x - 1

x−1 is a factor:

Answer:

11) Use identities to find the products:

Answer:

12) Factorise using identities:

Answer:

13) Expand using identities:

14) Factorise:

Answer:

15) Write the cubes in expanded form:

Answer:

16) Factorise:

Answer:

17) Verify the identity:

Answer:

18) Find the value without calculating cubes:

Answer:

19) Find possible expressions for dimensions of rectangles:

Answer:

20) Find possible expressions for dimensions of cuboids:

Answer:

FAQs: Class 9 Maths Chapter 2 – Polynomials

Share

Share

Company

Terms & Conditions

Resources

Teachers

2) Write the coefficient of $x^2$

$x^{2}$ in each of the following:

5) Find the value of the polynomial $5x - 4x^2 + 3$

$5 x - 4 x^{2} + 3$ :

8) Find the remainder when $x^3 + 3x^2 + 3x + 1$

$x^{3} + 3 x^{2} + 3 x + 1$ is divided by:

9) Determine which polynomial has $x + 1$

$x + 1$ as a factor:

10) Find the value of k if $x - 1$

$x - 1$ is a factor: