ALLEN Coaching
ApplyRegister for ALLEN Scholarship Test & get up to 90% Scholarship
Suppose you are buying 5 notebooks, each costing $x$ rupees, and 7 pencils, each costing $y$ rupees. Now, the total cost becomes $5x+7y$, which is an example of a polynomial. It helps us represent real-life scenarios in mathematical form. In the NCERT Exemplar Class 9 Chapter 2, you will find Polynomials, which are algebraic expressions of variables and coefficients - sound complex? Don't worry, we will break it down in a simple and logical manner that will help you build your foundation in algebra.
This article on NCERT Exemplar Class 9 Maths Solution Chapter 2, Polynomials, offers clear and step-by-step solutions for the exercise problems. These Polynomials class 9 ncert exemplar solutions are made by the Subject Matter Experts according to the latest CBSE syllabus, ensuring that students can grasp the basic concepts effectively. For the NCERT syllabus, books, notes, and class-wise solutions, refer to the NCERT.
NCERT Exemplar Class 9 Maths Solutions Chapter 2, Exercise: 2.1 Page: 14-16, Total Questions: 21 |
Which one of the following is a polynomial?
$(A) \frac{x^{2}}{2}-\frac{2}{x^{2}}\\ \\(B) \sqrt{2x}-1\\ \\(C)x^{2}+\frac{3x^{\frac{3}{2}}}{\sqrt{x}}\\ \\ (D)\frac{x-1}{x+2}$
Answer:
$x^{2}+\frac{3x^{\frac{3}{2}}}{\sqrt{x}}$Question:2
$\sqrt{2}$ is a polynomial of degree
(A) 2
(B) 0
(C) 1
(D) $\frac{1}{2}$
Answer:
[B]Question:3
Degree of the polynomial 4x4 + 0x3 + 0x5 + 5x + 7 is :
(A) 4
(B) 5
(C) 3
(D) 7
Answer:
[A]Question:4
Degree of the zero polynomial is
(A) 0
(B) 1
(C) Any natural number
(D) Not defined
Answer:
(D) Not definedQuestion:5
If $p(x)=x^{2}-2\sqrt{2}x+1$ , then $p\left (2\sqrt{2} \right )$ is equal to
(A) 0
(B) 1
(C) $\left (4\sqrt{2} \right )$
(D) $\left (8\sqrt{2} \right )$
Answer:
(B) 1Question:6
The value of the polynomial 5x – 4x2 + 3 when x = – 1 is
(A) –6
(B) 6
(C) 2
(D) –2
Answer:
[A]Question:7
If $p(x)=x+3$ , then $p(x)+p(-x)$ is equal to
(A) 3 (B) 2x (C) 0 (D) 6
Answer:
(D) 6Question:8
Zero of the zero polynomial is
(A) 0
(B) 1
(C) Any real number
(D) Not define
Answer:
(C) Any real numberQuestion:9
Zero of the polynomial $p(x)=2x+5$ is
(A) $-\frac{2}{5}$ (B) $-\frac{5}{2}$ (C) $\frac{2}{5}$ (D) $\frac{5}{2}$
Answer:
(B) $-\frac{5}{2}$Question:10
One of the zeroes of the polynomial $2x^{2}+7x-4$ is
(A) 2
(B) $\frac{1}{2}$
(C) $-\frac{1}{2}$
(D) -2
Answer:
(B) $\frac{1}{2}$Question:11
If $x^{51}+51$ is divided by x+1 , the remainder is
(A) 0
(B) 1
(C) 49
(D) 50
Answer:
(D) 50Question:12
If x+1 is a factor of the polynomial 2x2+kx , then the value of k is
(A) –3 (B) 4 (C) 2 (D) –2
Answer:
(C) 2Question:13
x+1 is a factor of the polynomial
$\\(A) x^{3}+x^{2}-x+1\\ (B) x^{3}+x^{2}+x+1\\ (C)x^{4}+x^{3}+x^{2}+1\\ (D)x^{4}+3x^{3}+3x^{2}+x+1$
Answer:
(B) $x^{3}+x^{2}+x+1$Question:14
One of the factors of $\left ( 25x^{2}-1 \right )+\left ( 1+5x \right )^{2}$ is
(A) 5+x (B) 5-x (C) 5x-1 (D) 10x
Answer:
(D) 10xQuestion:15
The value of $249^{2}-248^{2}$ is
(A) $1^{2}$ (B) 477 (C) 487 (D) 497
Answer:
(D) 497Question:16
The factorization of $4x^{2}+8x+3$ is
(A) $(x+1)(x+3)$
(B) $(2x+1)(2x+3)$
(C) $(2x+2)(2x+5)$
(D) $(2x-1)(2x-3)$
Answer:
Let us factorize the given polynomial $4x^{2}+8x+3$Question:17
Which of the following is a factor of $\left ( x+y \right )^{3}-\left ( x^{3}+y^{3} \right )$
$\\A. \ x^{2}+y^{2}+2xy\\ B. \ x^{2}+y^{2}-2xy\\ C. \ xy^{2}\\ D. \ 3xy$
Answer:
(D) 3xyQuestion:18
The coefficient of x in the expansion of (x+3)3 is
(A) 1 (B) 9 (C) 18 (D) 27
Answer:
(D) 27Question:19
If $\frac{x}{y}+\frac{y}{x}=-1$ $\left ( x,y\neq 0 \right )$ the vlaue of $x^{3}-y^{3}$ is
(A) 1 (B) -1 (C)0 (D) $\frac{1}{2}$
Answer:
(C) 0Question:20
If $49x^{2}-b=\left ( 7x + \frac{1}{2} \right )\left ( 7x - \frac{1}{2} \right )$ then the valueof b is,
$(A)0$ $(B)\frac{1}{\sqrt{2}}$ $(C)\frac{1}{4}$ $(D)\frac{1}{2}$
Answer:
Given: $49x^{2}-b=\left ( 7x + \frac{1}{2} \right )\left ( 7x - \frac{1}{2} \right )$Question:21
If $a+b+c=0$ , then $a^{3}+b^{3}+c^{3}$ is equal to
(A) 0 (B) abc (C) 3abc (D) 2abc
Answer:
(C) 3abcNCERT Exemplar Class 9 Maths Solutions Chapter 2, Exercise: 2.2 Page: 16-17, Total Questions: 2 |
Question:1
Which of the following expressions are polynomials? Justify your answer
(i) 8
(ii) $\sqrt{3}x^{2}-2x$
(iii) $1-\sqrt{5x}$
(iv) $\frac{1}{5x^{-2}}+5x+7$
(v) $\frac{(x-2)(x-4)}{x}$
(vi) $\frac{x}{x+1}$
(vii) $\frac{1}{7}a^{3}-\frac{2}{\sqrt{3}}a^{2}+4a-7$
(viii) $\frac{1}{2x}$
Answer:
(i, ii, iv, vii)Question:2
Write whether the following statements are True or False. Justify your answer.
i. A binomial can have at most two terms
ii. Every polynomial is a binomial.
iii. A binomial may have degree 5.
iv. Zero of a polynomial is always 0
v.A polynomial cannot have more than one zero.
vi. The degree of the sum of two polynomials each of degree 5 is always 5
Answer:
i. FalseNCERT Exemplar Class 9 Maths Solutions Chapter 2, Exercise: 2.3 Page: 18-22, Total Questions: 40 |
Question:1
Classify the following polynomials as polynomials in one variable, two variables etc.
(i) $x^{2}+x+1$
(ii) $y^{2}-5y$
(iii) $xy+yz+zx$
(iv) $x^{2}-2xy+y^{2}+1$
Answer:
(i) One VariableQuestion:2
Determine the degree of each of the following polynomials:
(i) $2x-1$
(ii) $-10$
(iii) $x^{3}-9x+3x^{5}$
(iv) $y^{3}(1-y^{4})$
Answer:
(i) OneQuestion:3
For the polynomial $\frac{x^{3}+2x+1}{5}-\frac{7}{2}x^{2}-x^{6}$, write
(i) The degree of the polynomial
(ii) The coefficient of $x^{3}$
(iii) The coefficient of $x^{6}$
(iv) The constant term
Answer:
(i) 6Question:4
Write the coefficient of $x^{2}$ in each of the following
(i) $\frac{\pi}{6}x+x^{2}-1$
(ii) $3x-5+0.x^{2}$
(iii) $(x-1)(3x-4)$
(iv) $(2x-5)(2x^{2}-3x+1)$
Answer:
(i) 1Question:5
Classify the following as a constant, linear, quadratic and cubic polynomials:
(i) $2-x^{2}+x^{3}$
(ii) $x^{3}$
(iii) $5t-\sqrt{7}$
(iv) $4-5y^{2}$
(v)$3.x^{0}$
(vi)$2+x$
(vii) $y^{3}-y$
(viii) $1+x+x^{2}$
(ix) $t^{2}$
(x) $\sqrt{2}x-1$
Answer:
(i) Cubic PolynomialQuestion:6
Give an example of a polynomial, which is:
(i) monomial of degree 1
(ii) binomial of degree 20
(iii) Trinomial of degree 2
Answer:
(i) xQuestion:7
Find the value of the polynomial $3x^{3} - 4x^{2} + 7x - 5$, when x = 3 and also when x = – 3.
Answer:
[61, –143]Question:8
If $p(x)=x^{2}-4x+3$ evaluate : $p(2)-p(-1)+p\left ( \frac{1}{2} \right )$
Answer:
$-\frac{31}{4}$Question:9
Find P(0), P(1), P(–2) for the following polynomials:
i. $P(x)=10x-4x^{2}-3$
ii. $P(y)=\left ( y+2 \right )\left ( y-2 \right )$
Answer:
(i) P(0) = –3; P(1) = 3; P(–2) = –39P(0) = – 4; P(1) = – 3; P(–2) = 0
Solution.
Given polynomial is P(y) = (y + 2) (y – 2)
Put y = 0,
P(0)= (0 + 2) (0 – 2)
= (2) (–2)
= – 4
Put y = 1,
P(1) = (1 + 2) (1 – 2)
= (3) (–1)
= – 3
Put y = – 2,
P(–2) = (–2 + 2) (–2 – 2)
= (0) (–4)
= 0
Hence, P(0) = – 4; P(1) = – 3; P(–2) = 0.
Question:10
Verify whether the following are True or False :
i)-3 is a zero of x-3
ii) $-\frac{1}{3}$ is a zero of $3x+1$
iii) $-\frac{4}{5}$ is a zero of $4-5y$
iv) 0 and 2 are the zeroes of $t^{2}-2t$
v) –3 is a zero of $y^{2}+y-6$
Answer:
i) FalseQuestion:11
Find the zeroes of the polynomial in each of the following :
i) p(x) = x - 4
ii) g(x) = 3 - 6x
iii)q(x) = 2x - 7
iv) h(y) = 2y
Answer:
i) 4Question:12
Find the zeroes of the polynomial : $p(x)=(x-2)^{2}-(x+2)^{2}$
Answer:
[0]Question:13
Answer:
Quotient $=x^{3}+x^{2}+x+1$Question:14
By Remainder Theorem find the remainder, when p(x) is divided by g(x) , where
$(i)p(x)=x^{3}-2x^{2}-4x-1; g(x)=x+1$
$(ii)p(x)=x^{3}-3x^{2}+4x+50; g(x)=x-3$
$(iii)p(x)=4x^{3}-12x^{2}+14x-3; g(x)=2x-1$
$(iv)p(x)=x^{3}-6x^{2}+2x-4; g(x)=1-\frac{3}{2}x$
Answer:
(i) 0Question:15
Check whether p(x) is multiple of g(x) or not
$\\(i)p(x)=x^{3}-5x^{2}+4x-3, g(x)=x-2$
$(ii) p(x)=2x^{3}-11x^{2}-4x+5, g(x)=2x+1$
Answer:
(i) NoQuestion:16
Show that
(i) $x+3$ is a factor of $69+11x-x^{2}+x^{3}$
(ii) $2x-3$ is a factor of $x+2x^{3}-9x^{2}+12$
Answer:
(i) Here $g(x)=x+3$ and $p(x)=69+11x-x^{2}+x^{3}$Question:17
Determine which of the following polynomials has $x-2a$ factor
(i) $3x^{2}+6x-24$
(ii) $4x^{2}+x-2$
Answer:
(i) $3x^{2}+6x-24$ onlyQuestion:18
Show that p - 1 is a factor of p10 - 1 and also of p11 - 1 .
Answer:
To prove : Here we have to prove that $p-1$ is a factor of $p^{10}-1$ and also of $p^{11}-1$ .Question:19
For what value of m is $x^{3}-2mx^{2}+16$ divisible by $x+2$ ?
Answer:
m = 1Question:20
If $x+2a$ is a factor of $x^{5}-4a^{2}x^{3}+2x+2a+3$ , find a.
Answer:
$a=\frac{3}{2}$Question:21
Find the value of m so that $2x-1$ be a factor of $8x^{4}+4x^{3}-16x^{2}+10x+m$ .
Answer:
m=-2Question:22
If $x+1$ is a factor of $ax^{3}+x^{2}-2x+4a-9$ , find the value of a.
Answer:
a = 2Question:23
Factorize :
(i) $x^{2}+9x+18$
(ii) $6x^{2}+7x-3$
(iii) $2x^{2}-7x-15$
(iv)$84-2r-2r^{2}$
Answer:
$(i)(x+6)(x+3)$Question:24
Factorize :
$(i) 2x^{3}-3x^{2}-17x+30$
$(ii) x^{3}-6x^{2}+11x-6$
$(iii) x^{3}+x^{2}-4x-4$
$(iv) 3x^{3}-x^{2}-3x+1$
Answer:
$(i) (x-2)(x+3)(2x-5)$Question:25
Using suitable identity, evaluate the following
$\\(i)103^{3}\\ (ii)101 \times 102\\ (iii)999^{2}$
Answer:
(i) 1092727Question:26
Factorize the following
$(i)4x^{2}+20x+25$
$(ii)9y^{2}-66yz+121z^{2}$
$(iii)\left ( 2x+\frac{1}{3} \right )^{2}-\left ( x-\frac{1}{2} \right )^{2}$
Answer:
$(i)\left ( 2x+5 \right )\left ( 2x+5 \right )$Question:27
Factorize the following :-
$(i) 9x^{2} - 12x + 3$
$(ii) 9x^{2} - 12x + 4$
Answer:
$(i)(x-1)(9x-3)$Question:28
Expand the following
$\\(i) (4a - b + 2c)^{2} \\ (ii) (3a - 5b - c)^{2}\\ (iii) (- x + 2y - 3z)^{2}$
Answer:
(i) $16a^{2}+b^{2}+4c^{2}-8ab-4bc+16ac$Question:29
Factorize the following
$(i)9x^{2}+4y^{2}+16z^{2}+12xy-16yz-24xz$
$(ii)25x^{2}+16y^{2}+4z^{2}-40xy+16yz-20xz$
$(iii)16x^{2}+4y^{2}+9z^{2}-16xy-12yz+24xz$
Answer:
$(i)(3x+2y-4z)(3x+2y-4z)$Question:30
If $a+b+c=9$ and $ab+bc+ca=26$ , find $a^{2}+b^{2}+c^{2}$ .
Answer:
29Question:31
Expand the following :
$(i)(3a-2b)^{3}$
$(ii)\left ( \frac{1}{x}+\frac{y}{3} \right )^{3}$
$(iii)\left ( 4-\frac{1}{3x} \right )^{3}$
Answer:
$(i)27a^{3}-8b^{2}-54a^{2}b+36ab^{2}$Question:32
Factorize the following :
$\\(i)1-64a^{3}-12a+48a^{2}\\ (ii)8p^{3}+\frac{12}{5}p^{2}+\frac{6}{25}p+\frac{1}{125}$
Answer:
(i)(1−4a)(1−4a)(1−4a)Question:33
Find the following product
(i) $\left (\frac{x}{2}+2y \right )\left ( \frac{x^{2}}{4}-xy+4y^{2} \right )$
(ii) $(x^{2}-1)(x^{4}+x^{2}+1)$
Answer:
(i) $\frac{x^{3}}{8}+8y^{3}$Question:34
Factorise:
(i) $1+64x^{3}$
(ii) $a^{3}-2\sqrt{2}b^{3}$
Answer:
(i) $(1+4x)(1+16x^{2}-4x)$Question:35
Find the following product $(2x-y+3z)(4x^{2}+y^{2}+9z^{2}+2xy+3yz-6xz)$
Answer:
$8x^{3}-y^{3}+27x^{3}$Question:36
Factorize
(i) $a^{3}-8b^{3}-64c^{3}-24abc$
(ii) $2\sqrt{2}a^{3}+8b^{3}-27c^{3}+18\sqrt{2}abc$
Answer:
(i) $(a-2b-4c)(a^{2}+4b^{2}+16c^{2}+2ab-8bc+4ac)$Question:37
Without actually calculating the cubes, find the value of
(i) $\left ( \frac{1}{2} \right )^{3}+\left ( \frac{1}{3} \right )^{3}-\left ( \frac{5}{6} \right )^{3}$
(ii) $(0.2)^{3}-(0.3)^{3}+(0.1)^{3}$
Answer:
(i) $\frac{-5}{12}$Question:38
Without finding the cubes, factorize $(x-2y)^{3}+(2y-3z)^{3}+(3z-x)^{3}$
Answer:
$3(x-2y)(2y-3z)(3z-x)$Question:39
Find the value of
(i) $x^{3}+y^{3}-12xy-64$ when $x+y=-4$
(ii) $x^{3}-8y^{3}-36xy-216$ when $x=2y+6$
Answer:
(i) 0Question:40
Answer:
Length (2a+3)NCERT Exemplar Class 9 Maths Solutions Chapter 2, Exercise: 2.4 Page: 23, Total Questions: 9 |
Question:1
Answer:
a = –1Question:2
Answer:
Value of a is 5Question:3
If both x – 2 and $x-\frac{1}{2}$ are factors of px2 + 5x + r, show that p = r.
Answer:
Let $f(x)=px^{2}+5x+r$Question:4
Answer:
First of all, factorize x2 – 3x + 2Question:5
Simplify: $(2x-5y)^{3}-(2x+5y)^{3}$
Answer:
$-250y^{2}-120x^{2}y$Question:6
Multiply: $x^{2}+4y^{2}+z^{2}+2xy+xz-2yz$ by $\left ( -z+x-2y \right )$
Answer:
$x^{3}-8y^{3}-z^{3}-6xyz$Question:7
If a, b, c are all non-zero and a + b + c = 0, prove that$\frac{a^{2}}{bc}+\frac{b^{2}}{ca}+\frac{c^{2}}{ab}=3$
Answer:
Given, $\frac{a^{2}}{bc}+\frac{b^{2}}{ca}+\frac{c^{2}}{ab}=3$Question:8
If a + b + c = 5 and ab + bc + ca = 10, then prove that a3 + b3 + c3 –3abc = – 25
Answer:
Given: (a + b + c) = 5, ab + bc + ca = 10Question:9
prove that: $(a+b+c)^{3}-a^{3}-b^{3}-c^{3}=3(a+b)(b+c)(c+a)$
Answer:
prove that: $(a+b+c)^{3}-a^{3}-b^{3}-c^{3}=3(a+b)(b+c)(c+a)$Topics covered in the NCERT exemplar Class 9 Maths solutions chapter 2 deal with the understanding of:
Given below are the subject-wise exemplar solutions of class 9 NCERT:
Here are the subject-wise links for the NCERT solutions of class 9:
Given below are the subject-wise NCERT Notes of class 9:
Here are some useful links for NCERT books and the NCERT syllabus for class 9:
A student can always solve the zeros of a polynomial with degree 2. It is known as the quadratic equation. Sometimes we can solve cubic equations or zeros of higher degree polynomials. With the help of a computer, we can draw the graph of a polynomial of any degree and then locate it's zero.
NCERT exemplar Class 9 Maths solutions chapter 2 states that it is the expansion of the nth power of any binomial such as (a+b)2; this is also known as a binomial theorem.
In an equation, two expressions are equated, whereas expression is the mathematical representation of different terms of any variable.
A clear understanding of Polynomials can prepare a student to solve problems based on Algebra, which ranges up to 3% of the whole paper.
Register for ALLEN Scholarship Test & get up to 90% Scholarship
Get up to 90% Scholarship on Offline NEET/JEE coaching from top Institutes
This ebook serves as a valuable study guide for NEET 2025 exam.
This e-book offers NEET PYQ and serves as an indispensable NEET study material.
As per latest 2024 syllabus. Physics formulas, equations, & laws of class 11 & 12th chapters
As per latest 2024 syllabus. Chemistry formulas, equations, & laws of class 11 & 12th chapters