NCERT Exemplar Class 8 Maths Chapter 7 Algebraic Expressions and Identities & Factorisation, presented here for students to prepare for exams. These exemplars solutions are designed by experts in accordance with CBSE syllabus(2018-2019), which covers all the topics of standard 8, Maths chapter 7.

This chapter is divided into two parts the first part of the chapter is about algebraic expressions and the second part is about factorization. In the first part, the students will learn about different types of algebraic expressions and their identities. In the second part, the students will learn how to factorize given equations. To learn the concepts in an easy way students are advised to solve the problems provided in the class 8 NCERT exemplar for chapter 7 algebraic expressions and identities & factorisation.

## Class 8 NCERT Exemplar For Algebraic Expressions and Identities & Factorisation

To understand the concepts present in each chapter of Maths as well as Science subject, solve NCERT exemplars for class 8, provided here. Let us discuss the topics based on which exemplars are given for chapter 7.

- Know about expressions, terms, factors, coefficient, Monomials, Binomials and Polynomials, Like and Unlike Term
- Operations performed on Algebraic Expression
- Identities of algebra and its applications
- Learn to find factors of natural numbers, algebraic expressions, by the method of common factors, by regrouping terms, using identities.
- Division of Algebraic Expressions
- Dividing monomial by monomial
- Dividing polynomial by monomial, etc.

Apart from these exemplars, students are also made available with exemplar books, NCERT solutions for 8th standard Maths and question papers to help students practice well for final exams. Students are advised to solve sample papers and previous year question papers which gives an idea of types of questions asked in the board exam from Algebraic expressions and identities and factorisation.

**NCERT Exemplar Class 8 Maths Solutions Chapter 7 Algebraic Expressions And Identities Factorisation:-**Download PDF Here

### Access NCERT Exemplar Class 8 Maths Chapter 7 Solutions

Exercise Page: 224

**In questions 1 to 33, there are four options out of which one is correct. Write the correct answer. **

**1. The product of a monomial and a binomial is a **

**(a) monomial (b) binomial **

**(c) trinomial (d) none of these**

**Solution:-**

(b) binomial

Let monomial = 2x, binomial = x + y

Then, product of a monomial and a binomial = (2x) Ã— (x + y)

= 2x^{2} + 2xy

**2. In a polynomial, the exponents of the variables are always **

**(a) integers (b) positive integers **

**(c) non-negative integers (d) non-positive integers**

**Solution:-**

(b) positive integers

**3. Which of the following is correct? **

**(a) (a â€“ b) ^{2} = a^{2} + 2ab â€“ b^{2} (b) (a â€“ b)^{2} = a^{2} â€“ 2ab + b^{2} **

**(c) (a â€“ b) ^{2} = a^{2} â€“ b^{2} (d) (a + b)^{2} = a^{2} + 2ab â€“ b^{2}**

**Solution:-**

(b) (a â€“ b)^{2} = a^{2} â€“ 2ab + b^{2}

We have, = (a – b) Ã— (a – b)

= a Ã— (a – b) â€“ b Ã— (a – b)

= a^{2} â€“ ab â€“ ba + b^{2}

= a^{2} â€“ 2ab + b^{2}

**4. The sum of â€“7pq and 2pq is **

**(a) â€“9pq (b) 9pq (c) 5pq (d) â€“ 5pq**

**Solution:-**

(d) â€“ 5pq

The given two monomials are like terms.

Then sum of -7pq and 2pg = – 7pq + 2pq

= (-7 + 2) pq

= -5pq

**5. If we subtract â€“3x ^{2}y^{2} from x^{2}y^{2}, then we get **

**(a) â€“ 4x ^{2}y^{2} (b) â€“ 2x^{2}y^{2} (c) 2x^{2}y^{2} (d) 4x^{2}y^{2}**

**Solution:-**

(d) 4x^{2}y^{2}

We have,

The given two monomials are like terms.

Subtract â€“3x^{2}y^{2} from x^{2}y^{2} = x^{2}y^{2} â€“ (- 3x^{2}y^{2})

= x^{2}y^{2} + 3x^{2}y^{2}

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

= 4x^{2}y^{2}

**6. Like term as 4m ^{3}n^{2} is **

**(a) 4m ^{2}n^{2} (b) â€“ 6m^{3}n^{2} (c) 6pm^{3}n^{2} (d) 4m^{3}n**

**Solution:-**

(b) â€“ 6m^{3}n^{2}

Like terms are formed from the same variables and the powers of these variables are also the same. But coefficients of like terms need not be the same.

**7. Which of the following is a binomial? **

**(a) 7 Ã— a + a (b) 6a ^{2} + 7b + 2c **

**(c) 4a Ã— 3b Ã— 2c (d) 6 (a ^{2} + b)**

**Solution:-**

(d) 6 (a^{2} + b)

Expressions that contain exactly two terms are called binomials.

= 6 (a^{2} + b)

= 6a^{2} + b

**8. Sum of a â€“ b + ab, b + c â€“ bc and c â€“ a â€“ ac is **

**(a) 2c + ab â€“ ac â€“ bc (b) 2c â€“ ab â€“ ac â€“ bc **

**(c) 2c + ab + ac + bc (d) 2c â€“ ab + ac + bc**

**Solution:-**

(a) 2c + ab â€“ ac â€“ bc

We have,

= (a â€“ b + ab) + (b + c â€“ bc) + (c â€“ a â€“ ac)

= a â€“ b + ab + b + c â€“ bc + c â€“ a â€“ ac

Now, grouping like terms

= (a – a) + (-b + b) + (c + c) + ab â€“ bc â€“ ac

= 2c + ab â€“ bc â€“ ac

**9. Product of the following monomials 4p, â€“ 7q ^{3}, â€“7pq is **

**(a) 196 p ^{2}q^{4} (b) 196 pq^{4} (c) â€“ 196 p^{2}q^{4} (d) 196 p^{2}q^{3}**

**Solution:-**

(a) 196 p^{2}q^{4}

= 4p Ã— (â€“ 7q^{3}) Ã— (â€“7pq)

= (4 Ã— (-7) Ã— (-7)) Ã— p Ã— q^{3} Ã— pq

= 196p^{2}q^{4}

**10. Area of a rectangle with length 4ab and breadth 6b ^{2} is **

**(a) 24a ^{2}b^{2} (b) 24ab^{3} (c) 24ab^{2} (d) 24ab**

**Solution:-**

(b) 24ab^{3}

We know that, area of rectangle = length Ã— breadth

Given, length = 4ab, breadth = 6b^{2}

= 4ab Ã— 6b^{2}

= 24ab^{3}

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

**(a) 12a ^{3}bc^{2} (b) 12a^{3}bc (c) 12a^{2}bc (d) 2ab +3ac + 2ac**

**Solution:-**

(a) 12a^{3}bc^{2}

We know that, volume of cuboid = length Ã— breadth Ã— height

Given, length = 2ab, breadth = 3ac, height = 2ac

= 2ab Ã— 3ac Ã— 2ac

= (2 Ã— 3 Ã— 2) Ã— ab Ã— ac Ã— ac

= 12a^{3}bc^{2}

**12. Product of 6a ^{2} â€“ 7b + 5ab and 2ab is **

**(a) 12a ^{3}b â€“ 14ab^{2} + 10ab (b) 12a^{3}b â€“ 14ab^{2} + 10a^{2}b^{2} **

**(c) 6a ^{2} â€“ 7b + 7ab (d) 12a^{2}b â€“ 7ab^{2} + 10ab**

**Solution:-**

(b) 12a^{3}b â€“ 14ab^{2} + 10a^{2}b^{2}

Now we have find product of trinomial and monomial,

= (6a^{2} â€“ 7b + 5ab) Ã— 2ab

= (2ab Ã— 6a^{2}) â€“ (2ab Ã— 7b) + (2ab Ã— 5ab)

= 12a^{3}b â€“ 14ab^{2} + 10a^{2}b^{2}

**13. Square of 3x â€“ 4y is **

**(a) 9x ^{2} â€“ 16y^{2} (b) 6x^{2} â€“ 8y^{2} **

**(c) 9x ^{2} + 16y^{2} + 24xy (d) 9x^{2} + 16y^{2} â€“ 24xy**

**Solution:-**

(d) 9x^{2} + 16y^{2} â€“ 24xy

As per the condition in the question, (3x â€“ 4y)^{2}

The standard identity = (a – b)^{2} = a^{2} â€“ 2ab + b^{2}

Where, a = 3x, b = 4y

Then,

(3x â€“ 4y)^{2} = (3x)^{2} â€“ (2 Ã— 3x Ã— 4y) + (4y)^{2}

= 9x^{2} â€“ 24xy + 16y^{2}

**14. Which of the following are like terms? **

**(a) 5xyz ^{2}, â€“ 3xy^{2}z (b) â€“ 5xyz^{2}, 7xyz^{2} **

**(c) 5xyz ^{2}, 5x^{2}yz (d) 5xyz^{2}, x^{2}y^{2}z^{2}**

**Solution:-**

(b) â€“ 5xyz^{2}, 7xyz^{2}

Like terms are formed from the same variables and the powers of these variables are also the same. But coefficients of like terms need not be the same.

**15. Coefficient of y in the term â€“y/3 is **

**(a) â€“ 1 (b) â€“ 3 (c) -1/3 (d) 1/3**

**Solution:-**

(c) -1/3

-y/3 can also be written as y Ã— (-1/3)

So, Coefficient of y is -1/3

**16. a ^{2} â€“ b^{2} is equal to **

**(a) (a â€“ b) ^{2} (b) (a â€“ b) (a â€“ b) **

**(c) (a + b) (a â€“ b) (d) (a + b) (a + b)**

**Solution:-**

(c) (a + b) (a â€“ b)

(a^{2} â€“ b^{2}) = (a + b) (a – b) is one of the standard identity.

**17. Common factor of 17abc, 34ab ^{2}, 51a^{2}b is **

**(a) 17abc (b) 17ab (c) 17ac (d) 17a ^{2}b^{2}c**

**Solution:-**

(b) 17ab

The given factors can be written in expanded form as,

17abc = 17 Ã— a Ã— b Ã— c

34ab^{2} = 2 Ã— 17 Ã— a Ã— b Ã— b

51a^{2}b = 3 Ã— 17 Ã— a Ã— a Ã— b

So, common factors in the above is 17 Ã— a Ã— b

= 17ab

**18. Square of 9x â€“ 7xy is **

**(a) 81x ^{2} + 49x^{2}y^{2} (b) 81x^{2} â€“ 49x^{2}y^{2} **

**(c) 81x ^{2} + 49x^{2}y^{2} â€“126x^{2}y (d) 81x^{2} + 49x^{2}y^{2} â€“ 63x^{2}y**

**Solution:-**

(c) 81x^{2} + 49x^{2}y^{2} â€“126x^{2}y

As per the condition in the question, (9x â€“ 7xy)^{2}

The standard identity = (a – b)^{2} = a^{2} â€“ 2ab + b^{2}

Where, a = 9x, b = 7xy

Then,

(9x â€“ 7xy)^{2} = (9x)^{2} â€“ (2 Ã— 9x Ã— 7xy) + (7xy)^{2}

= 81x^{2} â€“ 126x^{2}y + 49x^{2}y^{2}

**19. Factorised form of 23xy â€“ 46x + 54y â€“ 108 is **

**(a) (23x + 54) (y â€“ 2) (b) (23x + 54y) (y â€“ 2) **

**(c) (23xy + 54y) (â€“ 46x â€“ 108) (d) (23x + 54) (y + 2)**

**Solution:-**

(a) (23x + 54) (y â€“ 2)

Factorised form of 23xy â€“ 46x + 54y â€“ 108 is = 23xy â€“ (2 Ã— 23x) + 54y â€“ (2 Ã— 54)

Take out the common factors,

= 23x (y – 2) + 54 (y – 2)

Again take out the common factor,

= (y â€“ 2) (23x + 54)

**20. Factorised form of r ^{2} â€“ 10r + 21 is **

**(a) (r â€“ 1) (r â€“ 4) (b) (r â€“ 7) (r â€“ 3) **

**(c) (r â€“ 7) (r + 3) (d) (r + 7) (r + 3)**

**Solution:-**

(b) (r â€“ 7) (r â€“ 3)

Factorised form of r^{2} â€“ 10r + 21 is = r^{2} â€“ 7r â€“ 3r + 21

Take out the common factors,

= r (r – 7) – 3 (r – 7)

Again take out the common factor,

= (r â€“ 7) (r – 3)

**21. Factorised form of p ^{2} â€“ 17p â€“ 38 is **

**(a) (p â€“ 19) (p + 2) (b) (p â€“ 19) (p â€“ 2) **

**(c) (p + 19) (p + 2) (d) (p + 19) (p â€“ 2)**

**Solution:-**

(a) (p â€“ 19) (p + 2)

Factorised form of p^{2} â€“ 17p – 38 is = p^{2} â€“ 19p + 2p – 38

Take out the common factors,

= p (p – 19) + 2 (p – 19)

Again take out the common factor,

= (p â€“ 19) (p + 2)

**22. On dividing 57p ^{2}qr by 114pq, we get**

**(a) Â¼pr (b) Â¾pr (c) Â½pr (d) 2pr**

**Solution:-**

(c) Â½pr

On dividing 57p^{2}qr by 114pq,

It can be expanded as = (57 Ã— p Ã— p Ã— q Ã— r)/(114 Ã— p Ã— q)

= 57pr/114 â€¦ [divide both numerator and denominator by 57]

= Â½pr

**23. On dividing p (4p ^{2} â€“ 16) by 4p (p â€“ 2), we get **

**(a) 2p + 4 (b) 2p â€“ 4 (c) p + 2 (d) p â€“ 2**

**Solution:-**

(c) p + 2

On dividing p (4p^{2} â€“ 16) by 4p (p â€“ 2)

= (p((2p)^{2} â€“ (4)^{2}))/ (4p(p – 2))

= ((2p – 4) Ã— (2p + 4))/(4(p – 2))

Take out the common factors

= ((2(p – 2)) Ã— (2 (p + 4)))/(4(p -2))

= (4(p – 2)(p + 2))/ (4(p – 2))

= p + 2

**24. The common factor of 3ab and 2cd is **

**(a) 1 (b) â€“ 1 (c) a (d) c**

**Solution:-**

(a) 1

Considering the two monomials 3ab and 2cd there is no common factor except 1.

**25. An irreducible factor of 24x ^{2}y^{2} is **

**(a) x ^{2} (b) y^{2} (c) x (d) 24x**

**Solution:-**

(c) x

An irreducible factor is a factor which cannot be expressed further as a product of factors. Such a factorisation is called an irreducible factorisation.

24x^{2}y^{2} = 2 Ã— 2 Ã— 2 Ã— 3 Ã— x Ã— x Ã— y Ã— y

Therefore an irreducible factor is x.

**26. Number of factors of (a + b) ^{2} is **

**(a) 4 (b) 3 (c) 2 (d) 1**

**Solution:-**

(c) 2

Number of factors of (a + b)^{2} is = (a + b) (a + b) no further factorisation is possible.

**27. The factorised form of 3x â€“ 24 is **

**(a) 3x Ã— 24 (b) 3 (x â€“ 8) (c) 24 (x â€“ 3) (d) 3(x â€“ 12)**

**Solution:-**

(b) 3 (x â€“ 8)

The factorised form of 3x â€“ 24 is,

Take out 3 as common,

= 3 (x – 8)

**28. The factors of x ^{2} â€“ 4 are **

**(a) (x â€“ 2), (x â€“ 2) (b) (x + 2), (x â€“ 2) **

**(c) (x + 2), (x + 2) (d) (x â€“ 4), (x â€“ 4)**

**Solution:-**

(b) (x + 2), (x â€“ 2)

The factors of x^{2} â€“ 4 are,

X^{2} â€“ 4 = x^{2} – 2^{2}

= (x + 2) (x – 2)

**29. The value of (â€“ 27x ^{2}y) Ã· (â€“ 9xy) is **

**(a) 3xy (b) â€“ 3xy (c) â€“ 3x (d) 3x**

**Solution:-**

(d) 3x

The value of (â€“ 27x^{2}y) Ã· (â€“ 9xy) = (-27 Ã— x Ã— x Ã— y)/(- 9 Ã— x Ã— y)

= (27/9)x â€¦ [divide both numerator, denominator by 3]

** **= 3x

**30. The value of (2x ^{2} + 4) Ã· 2 is **

**(a) 2x ^{2} + 2 (b) x^{2} + 2 (c) x^{2} + 4 (d) 2x^{2} + 4**

**Solution:-**

(b) x^{2} + 2

The value of (2x^{2} + 4) Ã· 2 = (2x^{2} + 4)/2

= (2(x^{2} + 2))/2

= x^{2} + 2

**31. The value of (3x ^{3} +9x^{2} + 27x) Ã· 3x is **

**(a) x ^{2} +9 + 27x (b) 3x^{3} +3x^{2} + 27x **

**(c) 3x ^{3} +9x^{2} + 9 (d) x^{2} +3x + 9**

**Solution:-**

(d) x^{2} +3x + 9

The value of (3x^{3} +9x^{2} + 27x) Ã· 3x = (3x^{3} + 9x^{2} + 27x)/3x

Takeout 3x as common,

= 3x (x^{2} + 3x + 9)/3x

= x^{2} + 3x + 9

**32. The value of (a + b) ^{2} + (a â€“ b)^{2} is **

**(a) 2a + 2b (b) 2a â€“ 2b (c) 2a ^{2} + 2b^{2} (d) 2a^{2} â€“ 2b^{2}**

**Solution:-**

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

(a + b)^{2} + (a – b)^{2} = (a^{2} + b^{2} + 2ab) + (a^{2} + b^{2} â€“ 2ab)

= (a^{2} + a^{2}) + (b^{2} + b^{2}) + (2ab â€“ 2ab)

= 2a^{2} + 2b^{2}

**33. The value of (a + b) ^{2} â€“ (a â€“ b)^{2} is **

**(a) 4ab (b) â€“ 4ab (c) 2a ^{2} + 2b^{2} (d) 2a^{2} â€“ 2b^{2}**

**Solution:-**

(a) 4ab

The value of (a + b)^{2} â€“ (a â€“ b)^{2} = (a^{2} + b^{2} + 2ab) – (a^{2} + b^{2} â€“ 2ab)

= a^{2} – a^{2} + b^{2} – b^{2} + 2ab + 2ab

= 4ab

**In questions 34 to 58, fill in the blanks to make the statements true: **

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

**Solution:-**

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

Let us assume two like terms are, 3p and 2q

= 3p Ã— 2q

= 6pq

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

**Solution:-**

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

Let us assume two unlike terms are, – 3p and 2q

= -3p Ã— 2q

= – 6pq

**36. a (b + c) = a Ã— ____ + a Ã— _____. **

**Solution:-**

a (b + c) = a Ã— b + a Ã— c. â€¦ [by using left distributive law]

= ab + ac

**37. (a â€“ b) _________ = a ^{2} â€“ 2ab + b^{2}**

**Solution:-**

(a â€“ b) (a – b) = (a – b)^{2}= a^{2} â€“ 2ab + b^{2}

(a â€“ b) (a – b)= a Ã— (a – b) â€“ b Ã— (a – b)

= a^{2} â€“ ab â€“ ba + b^{2}

= a^{2} â€“ 2ab + b^{2}

**38. a ^{2} â€“ b^{2} = (a + b ) __________. **

**Solution:-**

a^{2} â€“ b^{2} = (a + b) (a – b) â€¦ [from the standard identities]

**39. (a â€“ b) ^{2} + ____________ = a^{2} â€“ b^{2}**

**Solution:-**

(a â€“ b)^{2} + (2ab â€“ 2b^{2}) = a^{2} â€“ b^{2}

= (a â€“ b)^{2} + (2ab â€“ 2b^{2})

= a^{2} + b^{2} â€“ 2ab + 2ab â€“ 2b^{2}

= a^{2} â€“ b^{2}

**40. (a + b) ^{2} â€“ 2ab = ___________ + ____________**

**Solution:-**

(a + b)^{2} â€“ 2ab = a^{2} + b^{2}

= (a + b)^{2} â€“ 2ab

= a^{2} + 2ab + b^{2} â€“ 2ab

= a^{2} + b^{2}

**41. (x + a) (x + b) = x ^{2} + (a + b) x + ________.**

**Solution:-**

(x + a) (x + b) = x^{2} + (a + b) x + ab

= (x + a) (x + b)

= x Ã— (x + b) + a Ã— (x + b)

= x^{2} + xb + xa + ab

= x^{2} + x (b + a) + ab

**42. The product of two polynomials is a ________. **

**Solution:-**

The product of two polynomials is a polynomials.

**43. Common factor of ax ^{2} + bx is __________. **

**Solution:-**

Common factor of ax^{2} + bx is x (ax + b)

**44. Factorised form of 18mn + 10mnp is ________. **

**Solution:-**

Factorised form of 18mn + 10mnp is 2mn (9 + 5p)

= (2 Ã— 9 Ã— m Ã— n) + (2 Ã— 5 Ã— m Ã— n Ã— p)

= 2mn (9 + 5p)

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