NCERT Exemplar solutions for Mathematics [English] Class 8 chapter 7 - Algebraic Expression, Identities and Factorisation [Latest edition]

The product of a monomial and a binomial is a ______.

In a polynomial, the exponents of the variables are always ______.

Which of the following is correct?

The sum of –7pq and 2pq is ______.

If we subtract –3x²y² from x²y², then we get ______.

Like term as 4m³n² is ______.

Which of the following is a binomial?

Sum of a – b + ab, b + c – bc and c – a – ac is ______.

Product of the following monomials 4p, – 7q³, –7pq is ______.

Area of a rectangle with length 4ab and breadth 6b² is ______.

Volume of a rectangular box (cuboid) with length = 2ab, breadth = 3ac and height = 2ac is ______.

Product of 6a² – 7b + 5ab and 2ab is ______.

Square of 3x – 4y is ______.

Which of the following are like terms?

Coefficient of y in the term `(-y)/3` is ______.

a² – b² is equal to ______.

Common factor of 17abc, 34ab², 51a²b is ______.

Square of 9x – 7xy is ______.

Factorised form of 23xy – 46x + 54y – 108 is ______.

Factorised form of r² – 10r + 21 is ______.

Factorised form of p² – 17p – 38 is ______.

On dividing 57p²qr by 114pq, we get ______.

On dividing p(4p² – 16) by 4p(p – 2), we get ______.

The common factor of 3ab and 2cd is ______.

An irreducible factor of 24x²y² is ______.

Number of factors of (a + b)² is ______.

The factorised form of 3x – 24 is ______.

The factors of x² – 4 are ______.

The value of (– 27x²y) ÷ (– 9xy) is ______.

The value of (2x² + 4) ÷ 2 is ______.

The value of (3x³ + 9x² + 27x) ÷ 3x is ______.

The value of (a + b)² + (a – b)² is ______.

The value of (a + b)² – (a – b)² is ______.

The product of two terms with like signs is a ______ term.

The product of two terms with unlike signs is a ______ term.

a(b + c) = a × ____ + a × _____.

(a – b) ______ = a² – 2ab + b² 

a² – b² = (a + b) ______.

(a – b)² + ______ = a² – b² 

(a + b)² – 2ab = ______ + ______.

(x + a)(x + b) = x² + (a + b)x + ______.

The product of two polynomials is a ______.

Common factor of ax² + bx is ______.

Factorised form of 18 mn + 10 mnp is ______.

Factorised form of 4y² – 12y + 9 is ______.

38x³y²z ÷ 19xy² is equal to ______.

Volume of a rectangular box with length 2x, breadth 3y and height 4z is ______.

67² – 37² = (67 – 37) × ______ = ______.

103² – 102² = ______ × (103 – 102) = ______.

Area of a rectangular plot with sides 4x² and 3y² is ______.

Volume of a rectangular box with l = b = h = 2x is ______.

The coefficient in – 37abc is ______.

Number of terms in the expression a² + bc × d is ______.

The sum of areas of two squares with sides 4a and 4b is ______.

The common factor method of factorisation for a polynomial is based on ______ property.

The side of the square of area 9y² is ______.

On simplification `(3x + 3)/3` = ______.

The factorisation of 2x + 4y is ______.

(a + b)² = a² + b² 

(a – b)² = a² – b² 

(a + b)(a – b) = a² – b² 

The product of two negative terms is a negative term.

The product of one negative and one positive term is a negative term.

The coefficient of the term – 6x²y² is – 6.

p²q + q²r + r²q is a binomial.

The factors of a² – 2ab + b² are (a + b) and (a + b).

h is a factor of 2π (h + r). 

Some of the factors of `n^2/2 + n/2` are `1/2, n` and (n + 1).

An equation is true for all values of its variables.

x² + (a + b)x + ab = (a + b)(x + ab)

Common factor of 11pq², 121p²q³, 1331p²q is 11p²q².

Common factor of 12a²b² + 4ab² – 32 is 4.

Factorisation of – 3a² + 3ab + 3ac is 3a(– a – b – c).

Factorised form of p² + 30p + 216 is (p + 18)(p – 12).

The difference of the squares of two consecutive numbers is their sum.

abc + bca + cab is a monomial.

On dividing `p/3` by `3/p`, the quotient is 9.

The value of p for 51² – 49² = 100p is 2.

(9x – 51) ÷ 9 is x – 51.

The value of (a + 1)(a – 1)(a² + 1) is a⁴ – 1.

Add:

7a²bc, –3abc², 3a²bc, 2abc² 

Add:

9ax + 3by – cz, –5by + ax + 3cz

Add:

xy²z² + 3x²y²z – 4x²yz², – 9x²y²z + 3xy²z² + x²yz² 

Add:

5x² – 3xy + 4y² – 9, 7y² + 5xy – 2x² + 13

Add:

2p⁴ – 3p³ + p² – 5p + 7, –3p⁴ – 7p³ – 3p² – p – 12

Add:

3a(a – b + c), 2b(a – b + c)

Add:

3a(2b + 5c), 3c(2a + 2b)

Subtract:

5a²b²c² from –7a²b²c²  

Subtract:

6x² – 4xy + 5y² from 8y² + 6xy – 3x² 

Subtract:

2ab²c² + 4a²b²c – 5a²bc² from –10a²b²c + 4ab²c² + 2a²bc² 

Subtract:

3t⁴ – 4t³ + 2t² – 6t + 6 from – 4t⁴ + 8t³ – 4t² – 2t + 11

Subtract:

2ab + 5bc – 7ac from 5ab – 2bc – 2ac + 10abc

Subtract:

7p(3q + 7p) from 8p(2p – 7q)

Subtract:

–3p² + 3pq + 3px from 3p(– p – a – r)

Multiply the following:

–7pq²r³, –13p³q²r

Multiply the following: 

3x²y²z², 17xyz

Multiply the following:

15xy², 17yz² 

Multiply the following: 

–5a²bc, 11ab, 13abc²

Multiply the following: 

–3x²y, (5y – xy)

Multiply the following: 

abc, (bc + ca)

Multiply the following: 

7pqr, (p – q + r)

Multiply the following: 

x²y²z², (xy – yz + zx)

Multiply the following: 

(p + 6), (q – 7)

Multiply the following: 

6mn, 0mn

Multiply the following: 

a, a⁵, a⁶ 

Multiply the following: 

–7st, –1, –13st² 

Multiply the following:

b³, 3b², 7ab⁵ 

Multiply the following: 

`- 100/9 rs; 3/4 r^3s^2`

Multiply the following: 

(a² – b²), (a² + b²) 

Multiply the following: 

(ab + c), (ab + c)

Multiply the following: 

(pq – 2r), (pq – 2r)

Multiply the following:  

`(3/4x - 4/3 y), (2/3x + 3/2y)`

Multiply the following: 

`3/2 p^2 + 2/3 q^2, (2p^2 - 3q^2)`

Multiply the following:  

(x² – 5x + 6), (2x + 7)

Multiply the following:

(3x² + 4x – 8), (2x² – 4x + 3)

Multiply the following:

(2x – 2y – 3), (x + y + 5)

Simplify:

(3x + 2y)² + (3x – 2y)²

Simplify:

(3x + 2y)² – (3x – 2y)²

Simplify: 

`(7/9 a + 9/7 b)^2 - ab`

Simplify:

`(3/4x - 4/3y)^2 + 2xy`

Simplify:

(1.5p + 1.2q)² – (1.5p – 1.2q)²

Simplify:

(2.5m + 1.5q)² + (2.5m – 1.5q)²

Simplify: 

(x² – 4) + (x² + 4) + 16

Simplify:

(ab – c)² + 2abc

Simplify:

(a – b) (a² + b² + ab) – (a + b) (a² + b² – ab)

Simplify:

(b² – 49)(b + 7) + 343

Simplify:

(4.5a + 1.5b)² + (4.5b + 1.5a)²

Simplify:

(pq – qr)² + 4pq²r

Simplify:

(s²t + tq²)² – (2stq)²

Expand the following, using suitable identities.

(xy + yz)²

Expand the following, using suitable identities.

(x²y – xy²)²

Expand the following, using suitable identities.

`(4/5a + 5/4b)^2`

Expand the following, using suitable identities.

`(2/3x - 3/2y)^2`

Expand the following, using suitable identities.

`(4/5p + 5/3q)^2`

Expand the following, using suitable identities.

(x + 3)(x + 7)

Expand the following, using suitable identities.

(2x + 9)(2x – 7)

Expand the following, using suitable identities.

`((4x)/5 + y/4)((4x)/5 + (3y)/4)`

Expand the following, using suitable identities.

`((2x)/3 - 2/3)((2x)/3 + (2a)/3)`

Expand the following, using suitable identities.

(2x – 5y)(2x – 5y)

Expand the following, using suitable identities.

`((2a)/3 + b/3)((2a)/3 - b/3)`

Expand the following, using suitable identities.

(x² + y²)(x² – y²)

Expand the following, using suitable identities.

(a² + b²)²

Expand the following, using suitable identities.

(7x + 5)²

Expand the following, using suitable identities.

(0.9p – 0.5q)²

Expand the following, using suitable identities.

x²y² = (xy)²

Using suitable identities, evaluate the following.

(52)² 

Using suitable identities, evaluate the following.

(49)² 

Using suitable identities, evaluate the following.

(103)² 

Using suitable identities, evaluate the following.

(98)² 

Using suitable identities, evaluate the following.

(1005)² 

Using suitable identities, evaluate the following.

(995)² 

Using suitable identities, evaluate the following.

47 × 53

Using suitable identities, evaluate the following.

52 × 53

Using suitable identities, evaluate the following.

105 × 95

Using suitable identities, evaluate the following.

104 × 97

Using suitable identities, evaluate the following.

101 × 103

Using suitable identities, evaluate the following.

98 × 103

Using suitable identities, evaluate the following.

(9.9)² 

Using suitable identities, evaluate the following.

9.8 × 10.2

Using suitable identities, evaluate the following.

10.1 × 10.2

Using suitable identities, evaluate the following.

(35.4)² – (14.6)²

Using suitable identities, evaluate the following.

(69.3)² – (30.7)²

Using suitable identities, evaluate the following.

(9.7)² – (0.3)²

Using suitable identities, evaluate the following.

(132)² – (68)²

Using suitable identities, evaluate the following.

(339)² – (161)²

Using suitable identities, evaluate the following.

(729)² – (271)²

Write the greatest common factor in the following terms.

–18a², 108a

Write the greatest common factor in the following terms.

3x²y, 18xy², – 6xy

Write the greatest common factor in the following terms.

2xy, –y², 2x²y

Write the greatest common factor in the following terms.

l²m²n, lm²n², l²mn²

Write the greatest common factor in the following terms.

21pqr, –7p²q²r², 49p²qr

Write the greatest common factor in the following terms.

qrxy, pryz, rxyz

Write the greatest common factor in the following terms. 

3x³y²z, – 6xy³z², 12x²yz³ 

Write the greatest common factor in the following terms. 

63p²a²r²s, – 9pq²r²s², 15p²qr²s², – 60p²a²rs² 

Write the greatest common factor in the following terms. 

13x²y, 169xy

Write the greatest common factor in the following terms. 

11x², 12y²

Factorise the following expression.

6ab + 12bc

Factorise the following expression.

– xy – ay

Factorise the following expression.

ax³ – bx² + cx

Factorise the following expression.

l²m²n – lm²n² – l²mn²

Factorise the following expression.

3pqr – 6p²q²r² – 15r²

Factorise the following expression.

x³y² + x²y³ – xy⁴ + xy

Factorise the following expression.

4xy² – 10x²y + 16x²y² + 2xy

Factorise the following expression.

2a³ – 3a²b + 5ab² – ab

Factorise the following expression.

63p²q²r²s – 9pq²r²s² + 15p²qr²s² – 60p²q²rs²

Factorise the following expression.

24x²yz³ – 6xy³z² + 15x²y²z – 5xyz

Factorise the following expression.

a³ + a² + a + 1

Factorise the following expression.

lx + my + mx + ly

Factorise the following expression.

a³x – x⁴ + a²x² – ax³ 

Factorise the following expression.

2x² – 2y + 4xy – x

Factorise the following expression.

y² + 8zx – 2xy – 4yz

Factorise the following expression.

ax²y – bxyz – ax²z + bxy² 

Factorise the following expression.

a²b + a²c + ab + ac + b²c + c²b

Factorise the following expression.

2ax² + 4axy + 3bx² + 2ay² + 6bxy + 3by² 

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

x² + 6x + 9

Factorise the following, using the identity a² + 2ab + b² = (a + b)².

x² + 12x + 36

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

x² + 14x + 49

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

x² + 2x + 1

Factorise the following, using the identity a² + 2ab + b² = (a + b)².

4x² + 4x + 1 

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

a²x² + 2ax + 1

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

a²x² + 2abx + b² 

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

a²x² + 2abxy + b²y² 

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

4x² + 12x + 9

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

16x² + 40x + 25

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

9x² + 24x + 16

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

9x² + 30x + 25

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

2x³ + 24x² + 72x

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

a²x³ + 2abx² + b²x

Factorise the following, using the identity a² + 2ab + b² = (a + b)².

4x⁴ + 12x³ + 9x² 

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

`x^2/4 + 2x + 4`

Factorise the following, using the identity a² + 2ab + b² = (a + b)². 

`9x^2 + 2xy + y^2/9`

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

x² – 8x + 16

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

x² – 10x + 25

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

y² – 14y + 49

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

p² – 2p + 1

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

4a² – 4ab + b² 

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

p²y² – 2py + 1

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

a²y² – 2aby + b²

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

9x² – 12x + 4

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

4y² – 12y + 9

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

`x^2/4 - 2x + 4`

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

a²y³ – 2aby² + b²y

Factorise the following, using the identity a² – 2ab + b² = (a – b)².

`9y^2 - 4xy + (4x^2)/9`

Factorise the following.

x² + 15x + 26

Factorise the following.

x² + 9x + 20

Factorise the following.

y² + 18x + 65

Factorise the following.

p² + 14p + 13

Factorise the following.

y² + 4y – 21

Factorise the following.

y² – 2y – 15

Factorise the following.

18 + 11x + x² 

Factorise the following.

x² – 10x + 21

Factorise the following.

x² – 17x + 60

Factorise the following.

x² + 4x – 77

Factorise the following.

y² + 7y + 12

Factorise the following.

p² – 13p – 30

Factorise the following.

a² – 16p – 80

Factorise the following using the identity a² – b² = (a + b)(a – b).

x² – 9

Factorise the following using the identity a² – b² = (a + b)(a – b).

4x² – 25y²

Factorise the following using the identity a² – b² = (a + b)(a – b).

4x² – 49y²

Factorise the following using the identity a² – b² = (a + b)(a – b).

3a²b³ – 27a⁴b

Factorise the following using the identity a² – b² = (a + b)(a – b).

28ay² – 175ax²

Factorise the following using the identity a² – b² = (a + b)(a – b).

9x² – 1

Factorise the following using the identity a² – b² = (a + b)(a – b).

25ax² – 25a

Factorise the following using the identity a² – b² = (a + b)(a – b).

`x^2/9 - y^2/25`

Factorise the following using the identity a² – b² = (a + b)(a – b).

`(2p^2)/25 - 32q^2`

Factorise the following using the identity a² – b² = (a + b)(a – b).

49x² – 36y²

Factorise the following using the identity a² – b² = (a + b)(a – b).

`y^3 - y/9`

Factorise the following using the identity a² – b² = (a + b)(a – b).

`x^2/25 - 625`

Factorise the following using the identity a² – b² = (a + b)(a – b).

`x^2/8 - y^2/18`

Factorise the following using the identity a² – b² = (a + b)(a – b).

`(4x^2)/9 - (9y^2)/16`

Factorise the following using the identity a² – b² = (a + b)(a – b).

`(x^3y)/9 - (xy^3)/16`

Factorise the following using the identity a² – b² = (a + b)(a – b).

1331x³y – 11y³x

Factorise the following using the identity a² – b² = (a + b)(a – b).

`1/36a^2b^2 - 16/49b^2c^2`

Factorise the following using the identity a² – b² = (a + b)(a – b).

a⁴ – (a – b)⁴

Factorise the following using the identity a² – b² = (a + b)(a – b).

x⁴ – 1

Factorise the following using the identity a² – b² = (a + b)(a – b).

y⁴ – 625

Factorise the following using the identity a² – b² = (a + b)(a – b).

p⁵ – 16p

Factorise the following using the identity a² – b² = (a + b)(a – b).

16x⁴ – 81

Factorise the following using the identity a² – b² = (a + b)(a – b).

x⁴ – y⁴

Factorise the following using the identity a² – b² = (a + b)(a – b).

y⁴ – 81

Factorise the following using the identity a² – b² = (a + b)(a – b).

16x⁴ – 625y⁴

Factorise the following using the identity a² – b² = (a + b)(a – b).

(a – b)² – (b – c)²

Factorise the following using the identity a² – b² = (a + b)(a – b).

(x + y)⁴ – (x – y)⁴

Factorise the following using the identity a² – b² = (a + b)(a – b).

x⁴ – y⁴ + x² – y²

Factorise the following using the identity a² – b² = (a + b)(a – b).

8a³ – 2a

Factorise the following using the identity a² – b² = (a + b)(a – b).

`x^2 - y^2/100`

Factorise the following using the identity a² – b² = (a + b)(a – b).

9x² – (3y + z)²

The following expression is the area of a rectangle. Find the possible length and breadth of the rectangle.

x² – 6x + 8

The following expression is the area of a rectangle. Find the possible length and breadth of the rectangle.

x² – 3x + 2

The following expression is the area of a rectangle. Find the possible length and breadth of the rectangle.

x² – 7x + 10

The following expression is the area of a rectangle. Find the possible length and breadth of the rectangle.

x² + 19x – 20

The following expression is the area of a rectangle. Find the possible length and breadth of the rectangle.

x² + 9x + 20

Carry out the following division:

51x³y²z ÷ 17xyz

Carry out the following division:

76x³yz³ ÷ 19x²y²

Carry out the following division:

17ab²c³ ÷ (–abc²)

Carry out the following division:

–121p³q³r³ ÷ (–11xy²z³)

Perform the following division:

(3pqr – 6p²q²r²) ÷ 3pq

Perform the following division:

(ax³ – bx² + cx) ÷ (– dx)

Perform the following division:

(x³y³ + x²y³ – xy⁴ + xy) ÷ xy

Perform the following division:

(– qrxy + pryz – rxyz) ÷ (– xyz)

Factorise the expression and divide them as directed:

(x² – 22x + 117) ÷ (x – 13)

Factorise the expression and divide them as directed:

(x³ + x² – 132x) ÷ x(x – 11)

Factorise the expression and divide them as directed:

(2x³ – 12x² + 16x) ÷ (x – 2)(x – 4)

Factorise the expression and divide them as directed:

(9x² – 4) ÷ (3x + 2)

Factorise the expression and divide them as directed:

(3x² – 48) ÷ (x – 4)

Factorise the expressions and divide them as directed:

(x⁴ – 16) ÷ x³ + 2x² + 4x + 8

Factorise the expression and divide them as directed:

(3x⁴ – 1875) ÷ (3x² – 75)

The area of a square is given by 4x² + 12xy + 9y². Find the side of the square.

The area of a square is 9x² + 24xy + 16y². Find the side of the square.

The area of a rectangle is x² + 7x + 12. If its breadth is (x + 3), then find its length.

The curved surface area of a cylinder is 2π(y² – 7y + 12) and its radius is (y – 3). Find the height of the cylinder (C.S.A. of cylinder = 2πrh).

The area of a circle is given by the expression πx² + 6πx + 9π. Find the radius of the circle.

The sum of first n natural numbers is given by the expression `n^2/2 + n/2`. Factorise this expression.

The sum of (x + 5) observations is x⁴ – 625. Find the mean of the observations.

The height of a triangle is x⁴ + y⁴ and its base is 14xy. Find the area of the triangle.

The cost of a chocolate is Rs (x + y) and Rohit bought (x + y) chocolates. Find the total amount paid by him in terms of x. If x = 10, find the amount paid by him.

The base of a parallelogram is (2x + 3 units) and the corresponding height is (2x – 3 units). Find the area of the parallelogram in terms of x. What will be the area of parallelogram of x = 30 units?

The radius of a circle is 7ab – 7bc – 14ac. Find the circumference of the circle. `(pi = 22/7)`

If p + q = 12 and pq = 22, then find p² + q².

If a + b = 25 and a² + b² = 225, then find ab.

If x – y = 13 and xy = 28, then find x² + y².

If m – n = 16 and m² + n² = 400, then find mn.

If a² + b² = 74 and ab = 35, then find a + b.

Verify the following:

(ab + bc)(ab – bc) + (bc + ca)(bc – ca) + (ca + ab)(ca – ab) = 0

Verify the following:

(a + b + c)(a² + b² + c² – ab – bc – ca) = a³ + b³ + c³ – 3abc

Verify the following:

(p – q)(p² + pq + q²) = p³ – q³

Verify the following:

(m + n)(m² – mn + n²) = m³ + n³

Verify the following:

(a + b)(a + b)(a + b) = a³ + 3a²b + 3ab² + b³ 

Verify the following:

(a – b)(a – b)(a – b) = a³ – 3a²b + 3ab² – b³ 

Verify the following:

(a² – b²)(a² + b²) + (b² – c²)(b² + c²) + (c² – a²) + (c² + a²) = 0

Verify the following:

(5x + 8)² – 160x = (5x – 8)² 

Verify the following:

(7p – 13q)² + 364pq = (7p + 13q)²

Verify the following:

`((3p)/7 + 7/(6p))^2 - (3/7p  + 7/(6p))^2 = 2`

Find the value of a, if 8a = 35² – 27² 

Find the value of a, if 9a = 76² – 67²

Find the value of a, if pqa = (3p + q)² – (3p – q)²

Find the value of a, if pq²a = (4pq + 3q)² – (4pq – 3q)² 

What should be added to 4c(– a + b + c) to obtain 3a(a + b + c) – 2b(a – b + c)?

Subtract b(b² + b – 7) + 5 from 3b² – 8 and find the value of expression obtained for b = – 3.

If `x - 1/x = 7` then find the value of `x^2 + 1/x^2`.

Factorise `x^2 + 1/x^2 + 2 - 3x - 3/x`.

Factorise p⁴ + q⁴ + p²q².

Find the value of `(6.25 xx 6.25 - 1.75 xx 1.75)/(4.5)`

Find the value of `(198 xx 198 - 102 xx 102)/96`

The product of two expressions is x⁵ + x³ + x. If one of them is x² + x + 1, find the other.

Find the length of the side of the given square if area of the square is 625 square units and then find the value of x.

Take suitable number of cards given in the adjoining diagram [G(x × x) representing x², R(x × 1) representing x and Y(1 × 1) representing 1] to factorise the following expressions, by arranging the cards in the form of rectangles:

1. 2x² + 6x + 4
2. x² + 4x + 4.

Factorise 2x² + 6x + 4 by using the figure.

Calculate the area of figure.

The figure shows the dimensions of a wall having a window and a door of a room. Write an algebraic expression for the area of the wall to be painted.

Match the expressions of column I with that of column II: