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NCERT solutions Mathematics Class 9 chapter 2 Polynomials

Chapter 2 - Polynomials

Page 32

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

(i) 4x^2 - 3x + 7

(ii) y^2+sqrt2

(iii) 3sqrtt+tsqrt2

(iv) y+2/y

(v) x^10+y^3+t^50

Q 1 | Page 32

Write the coefficients of x2 in each of the following:-

(i) 2+x^2+x

(ii) 2-x^2+x^3

(iii) pi/2x^2+x

(iv) sqrt2x-1

Q 2 | Page 32

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

Q 3 | Page 32

Write the degree of each of the following polynomials:-

(i) 5x3 + 4x2 +7x

(ii) 4 - y2

(iii) 5t - sqrt7

(iv) 3

Q 4 | Page 32

Classify the following as linear, quadratic and cubic polynomials:-

(i) x2 + x

(ii) x – x3

(iii) y + y2 + 4

(iv) 1 + x

(v) 3t

(vi) r2

(vii) 7x3

Q 5 | Page 32

Pages 34 - 35

Find the value of the polynomial 5x – 4x2 + 3 at

(i) x = 0

(ii) x = –1

(iii) x = 2

Q 1 | Page 34

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

p(y) = y2 – y + 1

Q 2.1 | Page 34

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

p(t) = 2 + t + 2t2 – t3

Q 2.2 | Page 34

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

p(x) = x3

Q 2.3 | Page 34

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

p(x) = (x – 1) (x + 1)

Q 2.4 | Page 34

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = 3x + 1, x =-1/3

Q 3.1 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = 5x – π, x = 4/5

Q 3.2 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = x2 – 1, x = 1, –1

Q 3.3 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = (x + 1) (x – 2), x = – 1, 2

Q 3.4 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = x2, x = 0

Q 3.5 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = lx + m, x = – m/l

Q 3.6 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = 3x^2 – 1, x = -1/sqrt3,2/sqrt3

Q 3.7 | Page 35

Verify whether the following zeroes of the polynomial, indicated against them.

p(x) = 2x + 1, x = 1/2

Q 3.8 | Page 35

Find the zero of the polynomial in the following case:- p(x) = x + 5

Q 4.1 | Page 35

Find the zero of the polynomial in the following case:- p(x) = x – 5

Q 4.2 | Page 35

Find the zero of the polynomial in the following case:- p(x) = 3x – 2

Q 4.2 | Page 35

Find the zero of the polynomial in the following case:- p(x) = 2x + 5

Q 4.3 | Page 35

Find the zero of the polynomial in the following case:- p(x) = 3x

Q 4.3 | Page 35

Find the zero of the polynomial in the following case:-

p(x) = ax, a ≠ 0

Q 4.4 | Page 35

Find the zero of the polynomial in the following case:- p(x) = cx + d, c ≠ 0, c, d are real numbers.

Q 4.5 | Page 35

Page 40

Find the remainder when x3 + 3x2 + 3x + 1 is divided by x+1.

Q 1.1 | Page 40

Find the remainder when x3 + 3x2 + 3x + 1 is divided by x - 1/2

Q 1.2 | Page 40

Find the remainder when x3 + 3x2 + 3x + 1 is divided by x.

Q 1.3 | Page 40

Find the remainder when x3 + 3x2 + 3x + 1 is divided by x + π.

Q 1.4 | Page 40

Find the remainder when x3 + 3x2 + 3x + 1 is divided by 5 + 2x.

Q 1.5 | Page 40

Find the remainder when x3 – ax2 + 6x – a is divided by x – a.

Q 2 | Page 40

Check whether 7 + 3x is a factor of 3x3 + 7x.

Q 3 | Page 40

Page 44

Determine which of the following polynomials has (x + 1) a factor :-

(i) x3 + x2 + x + 1

(ii) x4 + x3 + x2 + x + 1

(iii) x4 + 3x3 + 3x2 + x + 1

(iv) x^3-x^2-(2+sqrt2)x+sqrt2

Q 1 | Page 44

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in the following case:-

p(x) = 2x3 + x2 – 2x – 1, g(x) = x + 1

Q 2.1 | Page 44

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in the following case:-

p(x) = x3 + 3x2 + 3x + 1, g(x) = x + 2

Q 2.2 | Page 44

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in the following case:-

p(x) = x3 − 4x2 + x + 6, g(x) = x − 3

Q 2.3 | Page 44

Find the value of k, if x – 1 is a factor of p(x) in the following case:- p(x) = x2 + x + k

Q 3.1 | Page 44

Find the value of k, if x – 1 is a factor of p(x) in the following case:-

p(x) = 2x^2+kx+sqrt2

Q 3.2 | Page 44

Find the value of k, if x – 1 is a factor of p(x) in the following case:- p(x) = kx^2 - sqrt2x +1

Q 3.3 | Page 44

Find the value of k, if x – 1 is a factor of p(x) in the following case:-

p(x) = kx2 – 3x + k

Q 3.4 | Page 44

Factorise :- 12x2 – 7x + 1

Q 4.1 | Page 44

Factorise :- 2x2 + 7x + 3

Q 4.2 | Page 44

Factorise :- 6x2 + 5x – 6

Q 4.3 | Page 44

Factorise :- 3x2 – x – 4

Q 4.4 | Page 44

Factorise :- x3 – 2x2 – x + 2

Q 5.1 | Page 44

Factorise :- x3 – 3x2 – 9x – 5

Q 5.2 | Page 44

Factorise :- x3 + 13x2 + 32x + 20

Q 5.3 | Page 44

Factorise :- 2y3 + y2 – 2y – 1

Q 5.4 | Page 44

Pages 48 - 50

Use suitable identities to find the following products :- (x + 4) (x + 10)

Q 1.1 | Page 48

Use suitable identities to find the following products :- (x + 8) (x – 10)

Q 1.2 | Page 48

Use suitable identities to find the following products :- (3x + 4) (3x – 5)

Q 1.3 | Page 48

Use suitable identities to find the following products :-

(y^2+3/2)(y^2-3/2)

Q 1.4 | Page 48

Use suitable identities to find the following products :- (3 – 2x) (3 + 2x)

Q 1.5 | Page 48

Evaluate the following product without multiplying directly:- 103 × 107

Q 2.1 | Page 48

Evaluate the following product without multiplying directly:- 95 × 96

Q 2.2 | Page 48

Evaluate the following product without multiplying directly:- 104 × 96

Q 2.3 | Page 48

Factorise the following using appropriate identities:- 9x2 + 6xy + y2

Q 3.1 | Page 48

Factorise the following using appropriate identities:- 4y2 – 4y + 1

Q 3.2 | Page 48

Factorise the following using appropriate identities:- x^2 - y^2/100

Q 3.3 | Page 48

Expand following, using suitable identities :- (x + 2y + 4z)

Q 4.1 | Page 49

Expand following, using suitable identities :- (2x – y + z)2

Q 4.2 | Page 49

Expand following, using suitable identities :- (–2x + 3y + 2z)2

Q 4.3 | Page 49

Expand following, using suitable identities :- (3a – 7b – c)2

Q 4.4 | Page 49

Expand following, using suitable identities :- (–2x + 5y – 3z)2

Q 4.5 | Page 49

Expand following, using suitable identities :-[1/4a-1/2b+1]^2

Q 4.6 | Page 49

Factorise:- 4x2 + 9y2 + 16z2 + 12xy – 24yz – 16xz

Q 5.1 | Page 49

Factorise:- 2x^2+y^2+8z^2-2sqrt2xy+4sqrt2yz-8xz

Q 5.2 | Page 49

Write the following cube in expanded form:- (2x + 1)3

Q 6.1 | Page 49

Write the following cube in expanded form:- (2a – 3b)3

Q 6.2 | Page 49

Write the following cube in expanded form:- [3/2x+1]^3

Q 6.3 | Page 49

Write the following cube in expanded form:- [x-2/3y]^3

Q 6.4 | Page 49

Evaluate the following using suitable identities:-

(99)3

Q 7.1 | Page 49

Evaluate the following using suitable identities:- (102)3

Q 7.2 | Page 49

Evaluate the following using suitable identities:- (998)3

Q 7.3 | Page 49

Factorise :- 8a3 + b3 + 12a2b + 6ab2

Q 8.1 | Page 49

Factorise :- 8a3 – b3 – 12a2b + 6ab2

Q 8.2 | Page 49

Factorise :- 27 – 125a3 – 135a + 225a2

Q 8.3 | Page 49

Factorise :- 64a3 – 27b3 – 144a2b + 108ab2

Q 8.4 | Page 49

Factorise :-

27p^3-1/216-9/2p^2+1/4p

Q 8.5 | Page 49

Verify :-  x3 + y3 = (x + y) (x2 – xy + y2)

Q 9.1 | Page 49

Verify :- x3 – y3 = (x – y) (x2 + xy + y2)

Q 9.2 | Page 49

Factorise :- 27y3 + 125z3

Q 10.1 | Page 49

Factorise :- 64m3 – 343n3

Q 10.2 | Page 49

Factorise :- 27x3 + y3 + z3 – 9xyz

Q 11 | Page 49

Verify that x^3+y^3+z^3-3xyz=1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]

Q 12 | Page 49

If x + y + z = 0, show that x3 + y3 + z3 = 3xyz.

Q 13 | Page 49

Without actually calculating the cubes, find the value of the following:- (–12)3 + (7)3 + (5)3

Q 14.1 | Page 49

Without actually calculating the cubes, find the value of the following:- (28)3 + (–15)3 + (–13)3

Q 14.2 | Page 49

Give possible expressions for the length and breadth of the following rectangle, in which their areas are given:-

 Area : 25a2 – 35a + 12
Q 15.1 | Page 49

Give possible expressions for the length and breadth of the following rectangle, in which their areas are given:-

 Area : 35y2 + 13y –12
Q 15.2 | Page 49

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

 Volume : 3x2 – 12x
Q 16.1 | Page 50

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

 Volume : 12ky2 + 8ky – 20k
Q 16.2 | Page 50

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