Ncert exemplar solutions mathematics class 9 Solutions for Chapter 2 polynomials

Exercises in Chapter 2 polynomials Grade 9

Polynomials- Exercise 2.1

Polynomials - Exercise 2.2

Polynomials - Exercise 2.3

Polynomials - Exercise 2.4

Questions in Polynomials- Exercise 2.1

If a + b + c =0, then a³+b³ + c³ is equal to

Zero of the polynomial p(x) = 2x + 5 is

If x⁵¹+51 is divided by x + 1, the remainder is..

If x + 1 is a factor of the polynomial 2x² + kx, then the value of k is

One of the factors of (25x² – 1) + (1 + 5x)² is

The value of 249² – 248² is

x + 1 is a factor of the polynomial

If $49x^2-b=\left(7x+\frac{1}{2}\right)\left(7x-\frac{1}{2}\right)$ , then the value of b is

Which of the following is a factor of (x+ y)³ – (x³ + y³)?

The coefficient of x in the expansion of (x + 3)³ is

If $\frac{x}{y}+\frac{y}{x}=-1(\text{ where }x,y\ne0)$ , the value of x³+y³ is

One of the zeroes of the polynomial 2x² + 7x –4 is

The factorization of 4x² + 8x+ 3 is

Which one of the following is a polynomial?

√2 is a polynomial of degree

What is the degree of the polynomial $4x^4+0x^3+0x^5+5x+7$ ?

Degree of the zero polynomial is

If p(x)= x² – 2√2x + 1, then p(2√2) is equal to

The value of the polynomial 5x – 4x² + 3, when x = – 1 is

If p(x) = x + 3, then p(x) + p(–x) is equal to

Zero of the zero polynomial is

Which one of the following is a polynomial ?

(a) $\frac{x^{2}}{2}-\frac{2}{x^{2}}$

(b) $\sqrt{2 x}-1$

(d) $\frac{x-1}{x+1}$

√2 is a polynomial of degree

(A) 2

(B) 0

(D) ½

Degree of the polynomial

4x4 +0x³ + 0x5 + 5x+ 7 is

(A) 4

(B) 5

(d)7

Degree of the zero polynomial is

(A) 0

(B) 1

(D) Not defined

If p(x)= x² – 2√2x + 1, then p(2√2) is equal to

(A) 0

(B) 1

(D) 8√2 +1

The value of the polynomial 5x – 4x² + 3, when x = – 1 is

(A) – 6

(B) 6

(D) – 2

If p(x) = x + 3, then p(x) + p(–x) is equal to

(A) 3

(B) 2x

(D) 6

Zero of the zero polynomial is

(A) 0

(B) 1

(D) Not defined

Zero of the polynomial p(x) = 2x + 5 is

(A) – 2/5

(B) – 5/2

(D) 5/2

One of the zeroes of the polynomial 2x² + 7x –4 is

(A) 2

(B) ½

(D) -2

Questions in Polynomials - Exercise 2.2

Write whether the following statement is true or false:

Every polynomial is a Binomial.

Find whether $\sqrt{3}x^2-2x$ is a polynomial?

Write whether the following statement is true or false:

A Binomial can have atmost two terms.

Find whether $\frac{1}{2\mathrm{x}}$ is a polynomial?

Find whether $\frac{1}{7}a^3-\frac{2}{\sqrt{3}}a^2+4a-7$ is a polynomial?

Find whether $\frac{1}{x+1}$ is a polynomial?

Find whether $1-\sqrt{5x}$ is a polynomial?

Find whether $\frac{1}{5x^{-2}}+5x+7$ is a polynomial?

Write whether the following statement is true or false:

The degree of the sum of two polynomials each of degree 5 is always 5.

Find whether $8$ is a polynomial?

Find whether $\frac{(x-2)(x-4)}{x}$ is a polynomial?

Write whether the following statement is true or false:

A binomial may have degree 5

Write whether the following statement is true or false:

Zero of a polynomial is always 0.

Write whether the following statement is true or false:

A polynomial cannot have more than one zero.

Which of the following expressions are polynomials? Justify your answer:

Questions in Polynomials - Exercise 2.3

Factorise $4x^2+20x+25$ .

Find the factorised form of $x^{2}+4 y^{2}+9 z^{2}-4 x y-12 y z+6 x$ .

Find the factorised form of $16a^2+b^2+4c^2-8ab-4bc+16ac$ .

Find the product of $\left(\frac{x}{2}+2y\right)\left(\frac{x^2}{4}-xy+4y^2\right)$ .

If a + b + c = 9 and ab + bc + ca = 26, find a² + b² +c².

Find the expanded form of $\left(4-\frac{1}{3 x}\right)^{3}$ .

Find the expanded form of $\left(\frac{1}{x}+\frac{y}{3}\right)^{3}$ .

Find the expanded form of $(3a-2b)^3$ .

Factorise $8 p^{3}+\frac{12}{5} p^{2}+\frac{6}{25} p+\frac{1}{125}$ .

Factorise $1-64a^3-12a+48a^2$ .

Find the product of $\left(x^{2}-1\right)\left(x^{4}+x^{2}+1\right)$ .

Factorise $a^3-8b^3-64c^3-24abc$ .

Find the factorisation of $1+64 x^{3}$ .

Find the product of $(2x-y+3z)\left(4x^2+y^2+9z^2+2xy+3yz-6xz\right)$ .

Without actually calculating the cubes, find the value of $(0.2)^{3}-(0.3)^{3}+(0.1)^{3}$ .

Factorise $2 \sqrt{2} a^{3}+8 b^{3}-27 c^{3}+18 \sqrt{2} a b c$ .

Without finding the cubes, factorise (x- 2y)³ + (2y – 3z)³ + (3z – x)³.

Find the value of x³ -8y³ -36xy-216,when x = 2y + 6.

Without actually calculating the cubes, find the value of $\left(\frac{1}{2}\right)^2+\left(\frac{1}{3}\right)^3-\left(\frac{5}{6}\right)^3$ .

Give possible expression for the length and breadth of the rectangle whose area is given by 4a² +4a – 3.

Factorise $9 y^{2}-66 y z+121 z^{2}$ .

Check whether p(𝑥) is a multiple of g(𝑥) or not:

p(𝑥) = 𝑥³ – 5𝑥² + 4𝑥 – 3, g(𝑥) = 𝑥 – 2

Using suitable identity, evaluate $103^3$ .

Classify wether the given polynomial as polynomials in one variable, two variables, three variables:

$x^{2}-2 x y+y^{2}+1$

Classify wether the given polynomial as polynomials in one variable, two variables, three variables:

$x^2+x+1$

Classify wether the given polynomial as polynomials in one variable, two variables, three variables:

$y^3-5y$

Classify wether the given polynomial as polynomials in one variable, two variables, three variables:

$xy + yz + zx$

Determine the degree of $-10$ .

Determine the degree of $2x-1$ .

Determine the degree of $y^3 (1 - y^4)$ .

Write the coefficient of x² in $\frac{\pi}{6}x+x^2-1$ .

Classify whether $3$ as a constant, linear, quadratic and cubic polynomial.

Determine the degree of $x^3 - 9x + 3x^5$ .

For the polynomial $\frac{x^3+2x+1}{5}-\frac{7}{2}x^2-x^6$ , the constant term.

For the polynomial $\frac{x^3+2x+1}{5}-\frac{7}{2}x^2-x^6$ , the coefficient of x⁶.

For the polynomial $\frac{x^3+2x+1}{5}-\frac{7}{2}x^2-x^6$ , write the degree of the polynomial.

For the polynomial $\frac{x^3+2x+1}{5}-\frac{7}{2}x^2-x^6$ , the coefficient of x³.

Write the coefficient of x² in $(2x - 5) (2x2 - 3x + 1)$ .

Write the coefficient of x² in $(x -1) (3x - 4)$ .

Classify wether $2-x^2+x^3$ as a constant, linear, quadratic and cubic polynomial.

Classify whether $3x^3$ as a constant, linear, quadratic and cubic polynomial.

Write the coefficient of x² in $3x - 5$ .

Classify whether $t^2$ as a constant, linear, quadratic and cubic polynomial.

Classify whether $5 t-\sqrt{7}$ as a constant, linear, quadratic and cubic polynomial.

Classify wether $4 - 5y^2$ as a constant, linear, quadratic and cubic polynomial.

Classify whether $2 + x$ **as a constant, linear, quadratic and cubic polynomial.**

If p(𝑥) =𝑥² – 4𝑥 + 3, evaluate: 𝑝(2)− 𝑝(−1) + 𝑝(½).

Find the corresponding example of trinomial of degree 2.

Find the corresponding example of monomial of degree 1.

Find the corresponding example of binomial of degree 20.

Find the value of the polynomial 3𝑥³ – 4𝑥² + 7𝑥 – 5, when x = -3.

Find the value of the polynomial 3𝑥³ – 4𝑥² + 7𝑥 – 5, when x = 3.

Find $p(-2)$ for $(𝑥)=10𝑥-4𝑥^2-3$ .

Find the zeroes of the polynomial in $p(x)=x-4$ .

By actual division, find the quotient when the first polynomial is divided by the second polynomial: x⁴ + 1; x –1.

Find the zeroes of the polynomial:

p(𝑥)= (𝑥 –2)²−(𝑥 + 2)²

Find $p(0)$ for $(𝑥)=10𝑥-4𝑥^2-3$ .

Verify whether the following is true or false:

–3 is a zero of y² + y – 6

Verify whether the following is true or false:

– 4/5 is a zero of 4 –5y

Verify whether the following is true or false:

0 and 2 are the zeroes of t² – 2t

Find $p(-2)$ for $(𝑦)=(y + 2) (y - 2)$ .

Find $p(0)$ for $(𝑦)=(y + 2) (y - 2)$ .

Find $p(1)$ for $(𝑦)=(y + 2) (y - 2)$ .

Verify whether the following is true or false:

– 1/3 is a zero of 3x + 1

Verify whether the following is true or false:

–3 is a zero of x – 3.

Find the zeroes of the polynomial in $h(y)=2y$ .

Find the zeroes of the polynomial in $g(x)=3-6x$ .

Find the zeroes of the polynomial in $q(x)=2x-7$ .

By actual division, find the remainder when the first polynomial is divided by the second polynomial: x⁴ + 1; x –1.

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where

$p(x)=x^3-3x^2+4x+50,g(x)=x-3$

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where

$p(x)=x^3-6x^2+2x-4,g(x)=1-\frac{3}{2}x$

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where

$p(x)=4x^3-12x^2+14x-3,g(x)=2x-1$

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where

$\mathrm{p}(x)=x^3-2x^2-4x-1,g(x)=x+1$

Find the value of x³ +y³ -12xy + 64,when x+y = -4.

Check whether p(𝑥) is a multiple of g(𝑥) or not:

p(𝑥)= 2𝑥³ – 11𝑥²− 4𝑥 + 5, 𝑔(𝑥)= 2𝑥 + 1

State whether $2𝑥-3$ is a factor of $𝑥 + 2𝑥^3 - 9𝑥^2 + 12$ .

State whether $𝑥+3$ is a factor of $69 + 11𝑥-𝑥^2 + 𝑥^3$ .

State whether $𝑥-2$ is a factor of $4𝑥^2 + 𝑥-2$ .

State whether $𝑥-2$ is a factor of $3𝑥^2 + 6𝑥-24$ .

State wether $p-1$ is a factor of $p^{11} – 1$ .

State wether $p-1$ is a factor of $p^{10} – 1$ .

For what value of m is 𝑥³ – 2𝑚𝑥² + 16 divisible by x + 2?

If 𝑥 + 2𝑎 is a factor of 𝑥⁵ – 4𝑎²𝑥³ + 2𝑥 + 2𝑎 + 3, find a.

Find the value of m, so that 2x -1 be a factor of 8x⁴ +4x³ -16x² +10x+07.

Find the factorization of $84-2r-2r^2$ .

If x +1 is a factor of ax³ +x² -2x+4a-9, then find the value of a.

Find the factorization of $2x^3-3x^2-17x+30$ .

Factorise $16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz$ .

Find the factorization of $6x^2 +7x -3$ .

Find the factorization of $2x^2 -7x.-15$ .

Find the factorization of $x^2+9x+18$ .

Find the factorization of $3x^3 - x^2 - 3x +1$ .

Find the factorization of $x^3 + x^2 - 4x - 4$ .

Find the factorization of $x^3 -6x^2 +11 x-6$ .

Factorise $9x^2 +4y^2 + 16z^2 +12xy-16yz -24xz$ .

Factorise $9x^2 -12x+ 3$ .

Using suitable identity, evaluate $999^2$ .

Using suitable identity, evaluate $101\times102$ .

Factorise $\left(2 x+\frac{1}{3}\right)^{2}-\left(x-\frac{1}{2}\right)^{2}$ .

Factorise $25x^2 + 16y^2 + 4z^2 - 40xy + 16yz - 20xz$ .

Factorise $9 y^{2}-66 y z+121 z^{2}$ .

Factorise $9x^2 -12xy + 4$ .

Factorise $4x^2+20x+25$ .

Find the factorised form of $9 a^{2}+25 b^{2}+c^{2}-30 a b+10 b c-6 a c$ .

Classify the following polynomials as polynomials in one variable, two variables etc.

(i) x2 + x

(ii) y3 – 5y

(iii) xy + yz + zx

(iv)x2 – 2xy + y2 + 1

Determine the degree of each of the following polynomials:

(i) 2x – 1

(ii) –10

(iii) x3 – 9x + 3x5

(iv) y3 (1 – y4)

For the polynomial

$\frac{x^{3}+2 x+1}{5}-\frac{7}{2} x^{2}-x^{6}$

, write

(i) the degree of thepolynomial

(ii) the coefficient ofx3

(iii) the coefficient ofx6 the constant term

Write the coefficient of x2 in each of the following:

(i) (π/6)x + x2 – 1

(ii) 3x – 5

(iii) (x –1) (3x – 4)

(iv) (2x – 5) (2x2 – 3x + 1)

Classify the following as a constant, linear, quadratic and cubic polynomials:

(i) 2 – x2 + x3

(ii) 3x3

(iii) 5t – √7

(iv) 4 – 5y2

(v) 3

(vi) 2 + x

(vii) y3 – y

(viii) 1 + x + x2

(ix) t2

(x) √2x – 1

Give an example of a polynomial, which is:

(i) monomial of degree1

(ii) binomial of degree20

(iii) trinomial of degree 2

Find the value of the polynomial 3𝑥3 – 4𝑥2 + 7𝑥 – 5,

when x = 3 and also when x = –3.

If p(𝑥) =𝑥2 – 4𝑥 + 3,

evaluate: 𝑝(2)− 𝑝(−1) + 𝑝(½).

Find p(0), p(1),𝑝(−2) for the following polynomials:

(i) (𝑥)=10𝑥−4𝑥2 –3

(ii) (𝑦)=(y + 2) (y – 2)

Verify whether the following are true or false:

(i) –3 is a zero of x – 3

(ii) – 1/3 is a zero of 3x + 1

(iii) – 4/5 is a zero of 4 –5y

(iv) 0 and 2 are the zeroes of t2 – 2t

(v)–3 is a zero of y2 + y – 6

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 Solution:

Find the zeroes of the polynomial:

p(𝑥)= (𝑥 –2)2−(𝑥 + 2)2

By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial:

x4 + 1; x –1

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where (i) p(𝑥) = 𝑥3 – 2𝑥2 – 4𝑥 – 1, g(𝑥) = 𝑥 + 1

(ii) p(𝑥) = 𝑥3 – 3𝑥2 + 4𝑥 + 50, g(𝑥) = 𝑥 – 3 (iii) p(𝑥) = 4𝑥3 – 12𝑥2 + 14𝑥 – 3, g(𝑥) = 2𝑥 – 1 (iv) p(𝑥) = 𝑥3 – 6𝑥2 + 2𝑥 – 4, g(𝑥) = 1 – 3/2 𝑥

Check whether p(𝑥) is a multiple of g(𝑥) or not:

(i) p(𝑥) = 𝑥3 – 5𝑥2 + 4𝑥 – 3, g(𝑥) = 𝑥 – 2

(ii) p(𝑥)= 2𝑥3 – 11𝑥2− 4𝑥 + 5, 𝑔(𝑥)= 2𝑥 + 1

Show that:

(i) 𝑥 + 3 is a factor of

69 + 11𝑥−𝑥2 + 𝑥3.

(ii) 2𝑥−3 is a factor of

𝑥 + 2𝑥3 – 9𝑥2 + 12

Determine which of the following polynomials has x – 2 a factor:

(i) 3𝑥2 + 6𝑥−24.

(ii) 4𝑥2 + 𝑥−2.

Show that p – 1 is a factor of p10 – 1 and also of p11 – 1.

For what value of m is 𝑥3 – 2𝑚𝑥2 + 16 divisible by x + 2?

If 𝑥 + 2𝑎 is a factor of 𝑥5 – 4𝑎2𝑥3 + 2𝑥+ 2𝑎 + 3, find a.

Questions in Polynomials - Exercise 2.4

State whether the given statement is true or false.

(a +b +c)³ -a³ -b³– c³ =3(a +b)(b +c)(c +a).

If bothx– 2 andx– ½ are factors of px²+ 5x+r, state whether p=r.

Multiply x² + 4y² + z² + 2xy + xz – 2yz by (-z + x-2y).

If a, b, c are all non-zero and a + b + c = 0, state wether $\frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}=3$ .

If a+b+c= 5 and ab+bc+ca =10, then state whether that a³ +b³ +c³– 3abc = -25.

Simplify (2x- 5y)³ – (2x+ 5y)³.

Without actual division, check if 2x⁴– 5x³+ 2x²– x + 2 is divisible by x²– 3x + 2.

If the polynomials az³ + 4z² + 3z – 4 and z³ – 4z + a leave the same remainder when divided by z – 3, find the value of a.

The polynomial p(x) = x⁴ – 2x³ + 3x² – ax + 3a – 7 when divided by x + 1 leaves the remainder 19. Find the remainder when p(x) is divided by x + 2.

If the polynomials az3 + 4z2 + 3z – 4 and z3 – 4z + leave the same remainder when divided by z – 3, find the value of a.

The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by x + 1 leaves the remainder 19. Find the values of a. Also find the remainder when p(x) is divided by x + 2.

If both x – 2 and x – ½ are factors of px2 + 5x + r, show that p = r.

Without actual division, prove that 2x4 – 5x3 + 2x2 – x + 2 is divisible by x2 – 3x + 2.