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NCERT Solutions for Class 8 Maths Chapter 14 Factorization

NCERT Solutions for Class 8 Maths Chapter 14 Factorization

Edited By Ramraj Saini | Updated on Mar 01, 2024 06:47 PM IST

Factorization Class 8 Questions And Answers provided here. These NCERT Solutions are created by expert team at craeers360 keeping in mind of the latest syllabus and pattern of CBSE 2023-23. In NCERT solutions for Class 8 Maths chapter 14 Factorization, you will be dealing with questions related to algebraic expressions and natural numbers. Important topics like methods of common factors, factorization using identities, factorization by regrouping terms, factors of the form (x + a) ( x + b), and division of algebraic expressions are covered in this chapter.

There are 4 exercises with 34 questions given in the NCERT textbook. All these questions are prepared in the solutions of NCERT for Class 8 Maths chapter 14 Factorization in a step-by-step manner. It will be easy for you to understand the concept. For a better understanding of the concept, there are some practice questions given after every topic. You will find solutions to these practice questions also in NCERT Solutions for Class 8 maths by clicking on the link.

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Factorization Class 8 Solutions - Important Formulae

Factorization: Factorization is the process of expressing an algebraic equation as a product of its components. These components can be numbers, variables, or algebraic expressions.

Irreducible Factor: An irreducible factor is a component that cannot be further factored into a product of factors.

Method to Do Factorization:

The common factor approach involves three steps:

Write each term of the statement as a product of irreducible elements.

Look for and separate the similar components.

Combine the remaining elements in each term using the distributive law.

The regrouping approach involves grouping terms in a way that brings out a common factor across the groups.

Common Factor Identity: Certain factorable expressions take the form of:

  • a2 + 2ab + b2 = (a + b)2

  • a2 - 2ab + b2 = (a - b)2

  • a2 - b2 = (a + b)(a - b)

  • x2 + (a + b)x + ab = (x + a)(x + b)

Dividing a Polynomial by a Monomial: When dividing a polynomial by a monomial, you can divide each term of the polynomial by the monomial or use the common factor technique.

Division of Algebraic Expressions: In division of algebraic expressions, you factor both the dividend and the divisor, then cancel common factors.

Division Formula: Dividend = Divisor × Quotient or Dividend = Divisor × Quotient + Remainder.

Free download NCERT Solutions for Class 8 Maths Chapter 14 Factorization for CBSE Exam.

Factorization Class 8 NCERT Solutions (Intext Questions and Exercise)

NCERT Solutions for Class 8 Maths Chapter 14 Factorization - Topic 14.2.1 Method Of Common Factor

Question:(i) Factorise:

12 x +36

Answer:

We have
12x = 2 \times 2 \times 3 \times x
36 = 2 \times 2 \times 3 \times 3

So, we have 2 \times 2 \times 3 common in both
Therefore,

12x + 36 = 2 \times 2 \times 3 (x + 3)

12x + 36 = 12(x + 3)

Question:(ii) Factorise : 22y-32z

Answer:

We have,
22y= 2 \times 11 \times y
33z = 3 \times 11 \times z
So, we have 11 common in both
Therefore,

22y - 33z = 11(2y - 3z)

Question:(iii) Factorise :

( iii) \: \: 14 pq + 35 pqr

Answer:

We have
14pq = 2 \times 7 \times p \times q
35pqr = 5 \times 7 \times p \times q \times r
So, we have

7 \times p \times q common in both
Therefore,

14pq + 35pqr =7pq (2 + 5r)

Class 8 maths chapter 14 question answer - Exercise: 14.1

Question:1(i) Find the common factors of the given terms.

(i) 12 x , 36

Answer:

We have
12x ={\color{Red} 2 \times 2 \times 3}\times x
36 = {\color{Red} 2 \times 2 \times 3}\times 3
So, the common factors between the two are

2\times2\times3=12

Question:1(ii) Find the common factors of the given terms

(ii) 2y , 22xy

Answer:

We have,
2y = {\color{Red} 2 \times y}
22xy = {\color{Red} 2} \times 11 \times x {\color{Red} \times y}
Therefore, the common factor between these two is 2y

Question:1(iii) Find the common factors of the given terms

(iii) 14 pq, 28 p^2 q^2

Answer:

We have,
14pq = {\color{Red} 2 \times 7 \times p \times q}
28p^2q^2 = 2 \times {\color{Red} 2 \times 7 \times p} \times p{\color{Red} \times q} \times q
Therefore, the common factor is

2\times7\times p\times q=14pq

Question:1(iv) Find the common factors of the given terms.

(iv) 2x , 3x ^2 , 4

Answer:

We have,
2x = 2 \times x
3x^2 = 3 \times x \times x
4 = 2 \times 2
Therefore, the common factor between these three is 1

Question:1(v) Find the common factors of the given terms

( v ) 6 abc , 24 ab^2 , 12 a^2 b

Answer:

We have,
6abc ={\color{Red} 2 \times 3 \times a \times b }\times c
24ab^2 = 2 \times 2\times {\color{Red} 2\times 3 \times a \times b} \times b
12a^2b = 2 \times {\color{Red} 2\times 3 \times a} \times a{\color{Red} \times b}
Therefore, the common factors is

2 \times 3 \times a \times b = 6ab

Question:1(vi) Find the common factors of the given terms

(vi)16 x ^ 3 , -4 x ^ 2 , 32 x

Answer:

We have,
16x^3 = 2 \times 2 \times {\color{Red} 2 \times 2 \times x} \times x \times x
4x^2 = {\color{Red} 2 \times 2 \times x} \times x
32x = 2 \times 2 \times 2 \times{\color{Red} 2 \times 2 \times x}
Therefore, the common factors is

2 \times 2 \times x = 4x

Question:1(vii) Find the common factors of the given terms

(vii) 10 pq , 20 qr , 30 rp

Answer:

We have,
10pq ={\color{DarkRed} 2 \times 5} \times p \times q
20qr = 2\times{\color{DarkRed} 2 \times 5 }\times q \times r
30rp ={\color{DarkRed} 2}\times 3{\color{DarkRed} \times 5} \times r \times p
Therefore, the common factors between these three is

2 \times 5 =10

Question:1(viii) Find the common factors of the given terms

(viii)3 x ^2 y^3 , 10 x ^3 y ^ 2 , 6 x^ 2 y^2 z

Answer:

We have,
3x^{2}y^{2} = 3 \times {\color{Red} x \times x \times y \times y}
10x^{3}y^{2} =2 \times 5 \times x \times {\color{Red} x\times x \times y \times y}
6x^{2}y^{2}z =2 \times 3 \times{\color{Red} x \times x \times y \times y} \times z
Therefore, the common factors between these three are x\times x\times y \times y = x^{2}y^{2}

Question:2(i) Factorise the following expressions

(i)7x -42

Answer:

We have,
7x = 7 \times x \\ 42=7\times 2 \times 3=7\times 6\\ 7x-42=7x-7\times 6=7(x-6)

Therefore, 7 is a common factor

Question:2(ii) Factorise the following expressions

(ii)6 p - 12 q

Answer:

We have,
6p = 2 \times 3 \times p
12q = 2 \times 2 \times 3 \times q
\therefore on factorization

6p -12q = (2\times 3 \times p) - (2\times 2 \times 3 \times q) = (2\times 3)(p-2q) = 6(p-2q)

Question:2(iii) Factorise the following expressions

(iii)7 a ^2 + 14 a

Answer:

We have,
7a^2 = 7 \times a \times a
14a = 2 \times 7 \times a
\therefore 7a^2+14a = (7\times a \times a)+(2 \times 7 \times a) = (7 \times a)(a+2)
= 7a(a+2)

Question:2(iv) Factorise the following expressions

(iv)-16 z + 20 z^3

Answer:

We have,
-16z = -1 \times 2 \times 2 \times 2 \times 2 \times z
20z^3 = 2 \times 2 \times 5 \times z \times z \times z
\therefore on factorization we get,
-16z+20z^3 = (-1 \times 2 \times 2 \times 2 \times 2 \times z)+(2 \times 2 \times 5 \times z \times z \times z )
= (2\times 2 \times z)(-1 \times 2 \times 2+ 5 \times z \times z )
= 4z(-4+5z^2 )

Question:2(v) Factorise the following expressions

20 l^2 m + 30 alm

Answer:

We have,
20l^2m = 2 \times 2 \times 5 \times l \times l \times m
30alm = 2 \times 3 \times 5 \times a \times l \times m
\therefore on factorization we get,
20l^2m+30alm =(2\times 2 \times 5 \times l \times l \times m) + (2 \times 3 \times 5 \times a \times l \times m)
=(2\times 5 \times l \times m)(2\times l + 3 \times a )
=10lm(2l+3a)

Question:2(vi) Factorise the following expressions

5 x^2 y - 15 xy^2

Answer:

We have,
5x^2y = 5 \times x\times x \times y
15xy^2 =3\times 5 \times x\times y \times y
\therefore on factorization we get,
5x^2y - 15xy^2 = (5 \times x \times x \times y ) - (3\times 5 \times x \times y \times y )
=(5\times x \times y) ( x - 3\times y )
=5xy (x-3y)

Question:2(vii) Factorise the following expressions

10 a ^2 - 15 b^2 +20 c^2

Answer:

We have,
10a^2 = 2 \times 5 \times a \times a
15b^2 = 3 \times 5 \times b \times b
20c^2 = 2\times 2 \times 5 \times c \times c
\therefore on factorization we get,
10a^2-15b^2+20c^2 = (2\times 5 \times a \times a)-(3\times 5 \times b \times b)+(2\times 2 \times 5 \times c \times c) =5 (2 \times a \times a-3 \times b \times b+2\times 2 \times c \times c)
=5(2a^2-3b^2+4c^2)

Question:2(viii) Factorise the following expressions

- 4 a ^2 + 4 ab - 4ca

Answer:

We have,
-4a^2 = -1\times 2 \times 2 \times a\times a
4ab = 2 \times 2 \times a\times b
4ca = 2 \times 2 \times c\times a
\therefore on factorization we get,
-4a^2+4ab-4ca = (-1 \times 2 \times 2 \times a\times a )+( 2 \times 2 \times a\times b )- (2 \times 2 \times c\times a)

=(2 \times 2 \times a) (-1 \times a + b - c)
= 4a(-a+b-c)

Question:2(ix) Factorise the following expressions

x^2 yz + xy^2 z + xyz^2

Answer:

We have,
x^2yz =x \times x \times y \times z
xy^2z =x \times y \times y \times z
xyz^2 =x \times y \times z \times z
Therefore, on factorization we get,
x^2yz+xy^2z+xyz^2 =(x \times x \times y \times z)+(x \times y \times y \times z)+(x \times y \times z \times z)

=( x \times y \times z)(x + y + z)
=xyz(x+y+z)

Question:2(x) Factorise the following expressions

a x^2 y + bxy^2 + cxyz

Answer:

We have,
ax^2y = a \times x \times x \times y
bxy^2 = b \times x \times y \times y
cxyz = c \times x \times y \times z
Therefore, on factorization we get,
ax^2y+bxy^2+cxyz = ( a \times x \times x \times y)+( b \times x \times y \times y)+(c \times x \times y \times z) = (x\times y)( a \times x+ b \times y+c \times z)

= xy(ax+by+cz)

Question:3(i) Factorise x ^ 2 + xy + 8 x + 8y

Answer:

We have,
x^2 = x \times x
xy = x \times y
8x = 8 \times x
8y = 8 \times y
Therefore, on factorization we get,
x^2+xy+8x+8y = (x \times x)+(x\times y )+(8 \times x)+(8 \times y)
= x(x +y )+8(x+ y)
= (x+8)(x+y)

Question:3(ii) Factorise

15 xy -6 x +5 y -2

Answer:

We have,
15xy = 3 \times 5 \times x \times y
6x = 2 \times 3 \times x
5y = 5 \times y
2 = 2
Therefore, on factorization we get,
15xy - 6x +5y-2 = (3\times 5 \times x \times y)-(2 \times 3 \times x)+(5\times y)-2
=(5 \times y)(3\times x + 1)-2(3\times x + 1)
=(5y-2)(3x+1)

Question:3(iii) Factorise

ax + bx - ay - by

Answer:

We have,
ax+bx-ay-by = a(x-y)-b(x-y)
=(a-b)(x-y)
Therefore, on factorization we get,
(a-b)(x-y)

Question:3(iv) Factorise

15 pq + 15 + 9q + 25p

Answer:

We have,
15pq + 15 + 9q + 25p = 5 p(3q + 5) + 3 (3q + 5)
= (3q + 5)(5p + 3)
Therefore, on factorization we get,
(3q + 5)(5p + 3)

Question:3(v) Factorise

z-7 + 7 xy - xyz

Answer:

We have,
z - 7 + 7xy - xyz = z(1 - xy) -7(1 - xy)
= (1 - xy)(z - 7)
Therefore, on factorization we get,
(1 - xy)(z - 7)

Class 8 maths chapter 14 NCERT solutions - Exercise: 14.2

Question:1(i) Factorise the following expressions

a ^ 2 + 8a + 16

Answer:

We have,
a^2 + 8a + 16 = a^2+ 4a + 4a + 16
= a(a + 4) + 4 (a+4)
= (a+4)(a+4) = (a+4)^{2}
Therefore,
a^2+8a+16 = (a+4)^2

Question:1(ii) Factorise the following expressions

p^2 -10 p + 25

Answer:

We have,
p^2 - 10p + 25 = p^2 - 5p - 5p + 25
= p(p - 5) -5 (p -5)
= (p - 5)(p - 5) = (p-5)^{2}
Therefore,
p^2-10p+25 =(p-5)^2

Question:1(iii) Factorise the following expressions

25 m ^2 + 30 m + 9

Answer:

We have,
25m^2 + 30m + 9 = 25m^2 + 15m + 15m + 9
= 5m (5m + 3) +3(5m + 3)
= (5m + 3) (5m + 3) = (5m+3)^{2}
Therefore,
25m^2+30m+9 = (5m+3)^2

Question:1(iv) Factorise the following expressions

49 y^2 + 84 yz + 36 z^2

Answer:

We have,
49 y^2 + 84 yz + 36 z^2 = 49y^2 + 42yz + 42yz + 36z^2
= 7y(7y + 6z) + 6z(7y + 6z)
= (7y + 6z)(7y + 6z) = (7y+ 6z)^{2}
Therefore,
49y^2+84yz+36z^2=(7y+6z)^2

Question:1(v) Factorise the following expressions

4 x^2 - 8x + 4

Answer:

We have,
4 x^2 - 8x + 4 = 4x^2 - 4x - 4x + 4
= 4x(x - 1) -4(x - 1)
= 4(x-1)(x-1) \\\ \ \ = 4(x-1)^{2}

Question:1(vi) Factorise the following expressions

121 b^2 - 88 bc + 16 c^2

Answer:

We have,
121 b^2 - 88 bc + 16 c^2 = 121b^2 - 44bc - 44bc + 16c^2
= 11b(11b - 4c) - 4c(11b - 4c)
= (11b-4c)(11b-4c) = (11b -4c)^{2}
Therefore,
121 b^2 - 88 bc + 16 c^2 = (11b -4c)^{2}

Question:1(vii) Factorise the following expressions

( l+m ) ^2 - 4lm

Answer:

We have,
( l+m ) ^2 - 4lm = l^{2} + 2ml + m^{2} - 4lm (using \ (a+b)^{2} = a^{2} + 2ab + b^{2})
= l^{2} - 2lm + m^{2}
= (l-m)^{2} (using \ (a-b)^{2} = a^{_2} -2ab + b^{2})

Question:1(viii) Factorise the following expressions

a ^4 +2 a ^2 b ^ 2 + b ^ 4

Answer:

We have,
a ^4 +2 a ^2 b ^ 2 + b ^ 4 = a^{4} + a^{2}b^{2} + a^{2}b^{2} + b^{4}
= a^{2}(a^{2 }+ b^{2}) + b^{2}(a^{2}+b^{2}) = (a^{2}+b^{2})(a^{2}+b^{2}) = (a^{2}+b^{2})^{2}

Question:2(i) Factorise :

4 p^2 - 9 q ^2

Answer:

This can be factorized as follows
4 p^2 - 9 q ^2 = (2p)^{2} - (3q)^{2} = (2p - 3q)(2p + 3q) (using \ (a)^{2} - (b)^{2} = (a-b)(a+b))

Question:2(ii) Factorise the following expressions

63 a ^2 - 112 b ^ 2

Answer:

We have,
63 a ^2 - 112 b ^ 2 = 7 (9a^{2} - 16b^{2}) = 7 ((3a)^{2} - (4b)^{2}) =7 (3a - 4b)(3a + 4b)
(using \ (a)^{2} - (b)^{2} = (a-b)(a+b))


Question:2(iii) Factorise

49 x^2 - 36

Answer:

This can be factorised as follows
49 x^2 - 36 = (7x)^{2} - (6)^{2} = (7x - 6)(7x + 6) (using \ (a)^{2} - (b)^{2} = (a-b)(a+b) )

Question:2(iv) Factorise

16 x^5 - 144 x ^ 3

Answer:

The given question can be factorised as follows
16 x^5 - 144 x ^ 3 = 16x^3(x^{2}- 9)
= 16x^3((x)^{2}- (3)^{2}) = 16x^3(x-3)(x+3) (using \ (a)^{2}- (b)^{2} = (a-b)(a+b))

Question:2(v) Factorise

(l+m) ^ 2 - ( l- m ) ^2

Answer:

We have,
(l+m) ^ 2 - ( l- m ) ^2 = [(l + m) - (l - m)][(l + m) + (l - m)] (using a^{2} - b^{2} = (a-b)(a+b) )
= (l + m - l + m)(l + m + l - m)
= (2m)(2l) = 4ml

Question:2(vi) Factorise

9 x ^2 y^2 - 16

Answer:

We have,
9 x ^2 y^2 - 16 = (3xy)^{2} -(4)^{2} (using (a)^{2} -(b)^{2} = (a-b) (a+b) )
= (3xy - 4 )(3xy + 4)

Question:2(vii) Factorise

( x ^2 -2xy + y^2 ) - z ^2

Answer:

We have,
( x ^2 -2xy + y^2 ) - z ^2 = (x-y)^{2} - z^{2} (using \ (a-b)^{2} = a^{2} -2ab + b^{2})
= (x - y - z)(x - y + z) (using \ (a)^{2} - (b)^{2} = (a -b ) (a+b))

Question:2(viii) Factorise

25 a ^2 -4 b ^2 + 28 bc - 49 c ^2

Answer:

We have,
25 a ^2 -4 b ^2 + 28 bc - 49 c ^2 = 25a^{2} - (2b-7c)^{2} (using \ (a-b)^{2} = a^{2} -2ab + b^{2})
= (5a)^{2} - (2b-7c)^{2} (using \ (a)^{2} - (b)^{2} = (a -b ) (a+b))
=(5a - (2b - 7c))(5a + (2b - 7c) )
= (5a - 2b + 7c)(5a + 2b - 7c )

Question:3(i) Factorise the following expressions

ax ^2 + bx

Answer:

We have,
ax^2 = a \times x \times x
bx = b \times x
Therefore,
ax ^2 + bx = (a \times x \times x) + (b \times x)
= x(a \times x + b)
= x(ax + b)

Question:3(ii) Factorise the following expressions

7p^2 + 21 q ^2

Answer:

We have,
7p^2 = 7 \times p \times p
21q^3 = 3 \times 7 \times q \times q
Therefore,
7p^2 + 21 q ^2 = (7 \times p \times p) + (3 \times 7 \times q \times q)
=7 (p^{2}+ 3q^{2})

Question:3(iii) Factorise the following expressions

2 x^3 + 2xy^2 + 2 xz ^2

Answer:

We have,
2x^3 = 2 \times x \times x \times x
2xy^2 = 2 \times x \times y \times y
2xz^2 = 2 \times x \times z \times z
Therefore,
2 x^3 + 2xy^2 + 2 xz ^2 = (2 \times x \times x \times x) + ( 2 \times x \times y \times y) + ( 2 \times x \times z \times z)
= (2 \times x) [(x \times x) + (y \times y ) + (z \times z)]
= 2x(x^2+y^2+z^2)

Question:3(iv) Factorise the following expressions

am^2 + bm ^2 + bn ^2 + an^2

Answer:

We have,
am^2 + bm ^2 + bn ^2 + an^2 = m^2(a + b) + n^2(a + b)
= (a + b) (m^{2 }+n^{2})

Question:3(v) Factorise the following expressions

( lm + l ) + m + 1

Answer:

We have,
( lm + l ) + m + 1 = lm + l + m + 1
= l(m + 1) +1(m + 1)
= (m + 1)(l + 1)

Question:3(vi) Factorise the following expressions

y ( y + z ) + 9 ( y + z )

Answer:

We have,
y ( y + z ) + 9 ( y + z )
Take ( y+z) common from this
Therefore,
y ( y + z ) + 9 ( y + z ) = (y + z)(y + 9)

Question:3(vii) Factorise the following expressions

5 y ^ 2 - 20 y - 8z + 2yz

Answer:

We have,
5 y ^ 2 - 20 y - 8z + 2yz = 5y(y - 4) + 2z(y - 4)
= (y - 4)(5y + 2z)
Therefore,
5 y ^ 2 - 20 y - 8z + 2yz = (y - 4)(5y + 2z)

Question:3(viii) Factorise

10 ab + 4a + 5b + 2

Answer:

We have,
10 ab + 4a + 5b + 2 = 2a(5b + 2) + 1(5b + 2)
= (5b + 2)(2a + 1)
Therefore,
10 ab + 4a + 5b + 2 = (5b + 2)(2a + 1)

Question:3(ix) Factorise the following expressions

6 xy - 4 y + 6 - 9 x

Answer:

We have,
6 xy - 4 y + 6 - 9 x = 2y(3x - 2) - 3 (3x - 2)
= (3x - 2)(2y - 3)
Therefore,
6 xy - 4 y + 6 - 9 x = (3x - 2)(2y - 3)

Question:4(i) Factorise a ^ 4 - b ^ 4

Answer:

We have,
a ^ 4 - b ^ 4 = (a^{2})^{2} - (b^{2})^{2} = (a^{2} - b^{2})(a^{2} + b^{2}) = (a-b)(a+b)(a^{2} + b^{2})
using \ (x^{2} - y^{2}) = (x-y)(x+y)

Question:4(ii) Factorise p ^ 4 - 81

Answer:

We have,
p ^ 4 - 81 =
(p^{2})^{2} - (9)^{2} = (p^{2} - 9)(p^{2}+9) \\ . \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = (p^{2}-(3)^{2})(p^{2}+9)\\ .\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = (p-3)(p+3)(p^{2}+9) using \ a^{2} - b^{2} = (a-b)(a+b)

Question:4(iii) Factorise x ^4 - ( y + z )^4

Answer:

We have,
x ^4 - ( y + z )^4 =
(x^{2})^{2} -((y+z)^{2})^{2} = (x^{2} - (y+z)^{2})(x^{2} +(y+z)^{2})\\ \Rightarrow (x-(y+z))(x+(y+z))(x^{2} +(y+z)^{2})
(using \ a^{2} -b^{2} = (a-b)(a+b))

Question:4(iv) Factorise x ^ 4 - ( x-z ) ^ 4

Answer:

We have,
x ^ 4 - ( x-z ) ^ 4 = (x^{2})^{2} - ((x-z)^{2})^{2} using \ a^{2}-b^{2} = (a-b)(a+b)
= (x^{2} - (x-z)^{2})(x^{2}+(x-z)^{2})
= (x+(x-z))(x - (x-z))(x^{2}+(x-z)^{2})
= (2x - z)(z) ( x^{2}+(x-z)^{2} )

Question:4(v) Factorise a ^ 4 - 2 a^2 b^2 + b ^ 4

Answer:

We have,
a ^ 4 - 2 a^2 b^2 + b ^ 4 = a^{4} - a^{2}b^{2} - a^{2}b^{2} + b^{4}
= a^{2}(a^{2} - b^{2}) - b^{2}(a^{2} - b^{2})
= (a^{2} - b^{2}) (a^{2}-b^{2}) using \ a^{2}-b^{2} = (a-b)(a+b)
= (a^{2} - b^{2})^{2}
= ((a - b)(a+b))^{2}
= (a - b)^{2}(a+b)^{2}

Question:5(i) Factorise the following expression

p^ 2 + 6 p + 8

Answer:

We have,
p^ 2 + 6 p + 8 = p^{2} + 2p + 4p + 8
= p(p + 2) + 4(p + 2)
=(p + 2)(p + 4)
Therefore,
p^ 2 + 6 p + 8 =(p + 2)(p + 4)

Question:5(ii) Factorise the following expression

q ^ 2 - 10 q + 21

Answer:

We have,
q ^ 2 - 10 q + 21 = q^{2} - 7q -3q + 21
= q(q - 7) -3(q - 7)
=(q - 7)(q - 3)
Therefore,
q ^ 2 - 10 q + 21 =(q - 7)(q - 3)

Question:5(iii) Factorise the following expression

p^2 + 6 p - 16

Answer:

We have,
p^2 + 6 p - 16 = p^{2} + 8p - 2p - 16
= p(p + 8) -2(p + 8)
=(p - 2)(p + 8)
Therefore,
p^2 + 6 p - 16 =(p - 2)(p + 8)

Class 8 factorization NCERT solutions - Topic 14.3.1 Division Of A Monomial By Another Monomial

Question:(i) Divide 24 xy^2 z^3 \: \: by \: \: 6 yz^2

Answer:

We have,
\frac{24xy^{2}z^{3}}{6yz^{2}} =\frac{2\times 2\times 2\times3\times y \times y \times z\times z\times z}{2\times 3 \times y \times z \times z}= 4xyz

Question:(ii) Divide 63 a ^ 2 b^ 4 c ^6 \: \: by \: \: 7 a ^2 b^ 2 c ^3

Answer:

We have,
\frac{63a^{2}b^{4}c^{6}}{7a^{2}b^{2}c^{3}}=\frac{3\times 3 \times 7 \times a \times a \times b \times b\times b^2 \times c \times c \times c \times c^3}{7a^{2}b^{2}c^{3}} = 9b^{2}c^{3}

NCERT Solutions for Class 8 Maths Chapter 14 Factorization-Exercise: 14.3

Question:1(i) Carry out the following divisions

28 x ^ 4 \div 56 x

Answer:

, \frac{28x^{4}}{56x} = \frac{2 \times 2 \times 7 \times x \times x \times x \times x}{2 \times 2 \times 2 \times 7 \times x} = \frac{x^{3}}{2}

This is done using factorization.

Question:1(ii) Carry out the following divisions

-36 y^3 \div 9 y^2

Answer:

We have,
-36 y^{3} = -1 \times 2 \times 2 \times 3 \times 3 \times y \times y \times y
9 y^{2 } = 3 \times 3 \times y \times y
Therefore,

\frac{-36y^{3}}{9y^{2}} = \frac {-1 \times 2 \times 2 \times 3 \times 3 \times y \times y \times y}{3 \times 3 \times y \times y} = -4y

Question:1(iii) Carry out the following divisions

66 pq^2 r ^ 3 \div 11 q r ^2

Answer:

We have,
66pq^2r^3 = 2 \times 3 \times 11 \times p \times q \times q \times r \times r \times r
11qr^2 = 11 \times q \times r \times r
Therefore,
\frac{66pq^{2}r^{3}}{11qr^{2}} = \frac{2 \times 3 \times 11 \times p \times q \times q \times r \times r \times r}{11 \times q \times r \times r} = 6pqr

Question:1(iv) Carry out the following divisions

34 x^ 3 y^3 z ^ 3 \div 51 x y^2 z ^ 3

Answer:

We have,

\therefore \frac{34x^{3}y^{3}z^{3}}{51xy^{2}z^{3}} = \frac{2 \times 17\times \ x \times x \times x \times y \times y \times y \times z\times z \times z}{3 \times 17 \times x \times y \times y \times z \times z\times z} = \frac{2x^{2}y}{3}

Question:1(v) Carry out the following divisions

12 a ^ 8 b^ 8 \div ( -6 a ^ 6 b ^ 4 )

Answer:

We have,

\frac{12a^8b^8}{-6a^4b^4}= \frac{2 \times 2 \times 3 \times a \times a \times a^{6} \times b \times b \times b \times b \times b^{4}}{-1 \times 2 \times 3 \times a^{6} \times b^{4}} = -2a^{2}b^{4}

Question:3(i) workout the following divisions

( 10 x - 25) \div 5

Answer:

We have,
10x -25 = 5(2x - 5)
Therefore,
\frac{10x-25}{5}= \frac{5(2x-5)}{5} = 2x - 5

Question:3(ii) workout the following divisions

( 10 x -25 ) \div ( 2x -5 )

Answer:

We have,
10x-25 = 5(2x - 5 )
Therefore,
\frac{10x-25}{2x-5} = \frac{5(2x-5)}{2x-5} = 5

Question:3(iii) workout the following divisions

10 y ( 6y +21 ) \div 5 ( 2y + 7 )

Answer:

We have,
10y(6y + 21) = 2 \times y \times 5 \times 3(2y + 7)
Therefore,
\frac{10y(6y+21)}{5(2y+7)} = \frac{2 \times 5 \times y \times 3(2y+7)}{5(2y+7)} = 6y

Question:3(iv) workout the following divisions

9 x ^2 y^2 ( 3z -24 ) \div 27 xy ( z-8 )

Answer:

We have,
9x^{2}y^{2}(3z-24) = 9x^{2}y^{2} \times 3(z-8) = 27x^{2}y^{2}(z-8)

\therefore \frac{9x^{2}y^{2}(3z-24)}{27xy(z-8)} = \frac{27x^{2}y^{2}(z-8)}{27xy(z-8)} = xy

Question:3(v) workout the following divisions

96 abc ( 3a -12 ) ( 5 b -30 ) \div 144 (a-4 ) ( b- 6 )

Answer:

We have,
96abc(3a - 12)(5b - 30) = 2 \times 48abc \times 3(a - 4) \times 5(b - 6)
= 2 \times144abc (a - 4) \times 5(b - 6)
Therefore,
\frac{96abc(3a-12)(5b-30)}{144(a-4)(b-6)} = \frac{2 \times 144abc (a-4) \times 5 (b - 6)}{144(a-4)(b-6)} = 10abc

Question:4(i) Divide as directed

5 ( 2x +1 ) ( 3x +5 ) \div ( 2x +1)

Answer:

We have,
\frac{5(2x+1)(3x+5)}{2x+1} = 5(3x+5)

Question:4(ii) Divide as directed

26 xy ( x+5 ) ( y-4) \div 13 x ( y-4 )

Answer:

We have,
\frac{26xy(x+5)(y-4)}{13x(y-4)} = \frac{2 \times 13xy(x+5)(y-4)}{13x(y-4)} =2y(x+5)

Question:4(iii) Divide as directed

52 pqr ( p+ q ) ( q+ r ) ( r +p)\div 104 pq ( q+r ) ( r + p )

Answer:

We have,
\frac{52pqr(p+q)(q+r)(r+p)}{104pq(q+r)(r+p)} = \frac{r(p+q)}{2}

Question:4(iv) Divide as directed

20 ( y+4 ) ( y^2 + 5 y + 3 ) \div 5 (y +4 )

Answer:

We have,
\frac{20(y+4)(y^{2}+5y+3)}{5(y+4)} =\frac{4 \times 5(y+4)(y^{2}+5y+3)}{5(y+4)} = 4(y^{2}+5y+3)

Question:4(v) Divide as directed

x ( x+1 ) ( x+2 ) ( x+3 ) \div x ( x+1 )

Answer:

We have,
\frac{x(x+1)(x+2)(x+3)}{x(x+1)} = (x+2)(x+3)

Question:5(iv) Factorise the expression and divide then as directed

4 yz ( z^2 + 6z -16 ) \div 2y ( z+8 )

Answer:

We first simplify our numerator
So,
4yz( z^2+ 6z - 16)
Add and subtract 64 \Rightarrow 4yz( z^2- 64 + 6z - 16 + 64)
= 4yz(z^2-8^2 + 6z + 48)
= 4yz((z + 8)(z - 8) + 6(z + 8)) using \ a^{2} -b^{2} = (a - b)(a + b)
= 4yz (z + 8)(z - 8 + 6)
= 4yz(z + 8)(z - 2)
Now,
\frac{4yz(z^{2}+6z-16)}{2y(z+8)} = \frac{4yz(z+8)(z-2)}{2y(z+8)}= 2z(z-2)

Question:5(vi) Factorise the expression and divide then as directed

12 xy ( 9 x^2 - 16 y^2 ) \div 4 xy ( 3 x + 4 y )

Answer:

We first simplify our numerator,
12xy ( 9x^{2} -16y^{2} ) = 12xy (3x)^{2} -(4y)^{2}

using (a)^{2} -(b)^{2} = (a-b)(a+b)
= 12xy((3x - 4y)(3x + 4y))
Now,
\frac{12xy(9x^{2} - 16y^{2})}{4xy(3x + 4y)} = \frac{12xy(3x+4y)(3x-4y)}{4xy(3x+4y)} = 3(3x-4y)

Question:5(vii) Factorise the expression and divide then as directed

39 y^2 ( 50 y^2 - 98 ) \div 26 y^2 ( 5y +7 )

Answer:

We first simplify our numerator,
39y^{2}(50y^{2} -98) = 39y^{2} \times 2(25y^{2} - 49) using (a)^{2} -(b)^{2} = (a-b)(a+b)
= 78y^{2} ((5y)^{2} - (7)^{2})
= 78y^{2} (5y - 7)(5y+7)
Now,
\frac{39y^{2}(50y^{2}-98)}{26y^{2}(5y +7)} = \frac{78y^{2}(5y-7)(5y+7)}{26y^{2}(5y+7)} = 3(5y-7)

NCERT Solutions for Class 8 Maths Chapter 14 Factorization-Exercise: 14.4

Question:1 Find and correct the errors in the following mathematical statements

4 ( x-5 ) = 4 x - 5

Answer:

Our L.H.S.
= 4(x - 5) = 4x - 20
R.H.S. = 4x -5
It is clear from the above that L.H.S. is not equal to R.H.S.
So, correct statement is
4(x - 5) = 4x - 20

Question:2 Find and correct the errors in the following mathematical statements

x ( 3 x + 2 ) = 3 x^2 +2

Answer:

Our L.H.S.
= x(3x + 2) = 3x^2 + 2x
R.H.S.= 3x^2 + 2
It is clear from the above that L.H.S. is not equal to R.H.S.
So, correct statement is
= x(3x + 2) = 3x^2 + 2x

Question:3 Find and correct the errors in the following mathematical statements

2 x + 3y = 5 xy

Answer:

Our L.H.S. = 2x + 3y
R.H.S. = 5xy
It is clear from the above that L.H.S. is not equal to R.H.S.
SO, correct statement is
2x + 3y = 2x + 3y

Question:4 Find and correct the errors in the following mathematical statements

x + 2x + 3x = 5x

Answer:

Our L.H.S. = x + 2x + 3x = 6x
R.H.S. = 5x
It is clear from the above that L.H.S. is not equal to R.H.S.
So, correct statement is
x + 2x + 3x = 6x

Question:5 Find and correct the errors in the following mathematical statements

(Q5)\ 5 y + 2y + y - 7y = 0

Answer:

Our L.H.S. is
5y + 2y + y - 7y = y
R.H.S. = 0
IT is clear from the above that L.H.S. is not equal to R.H.S.
So, Correct statement is
5y + 2y + y - 7y = y

Question:6 Find and correct the errors in the following mathematical statements

3 x +2 x = 5 x ^2

Answer:

Our L.H.S. is
3x + 2x = 5x
R.H.S. = 5x^2
It is clear from the above that L.H.S. is not equal to R.H.S.
So, correct statement is
3x + 2x = 5x

Question:7 Find and correct the errors in the following mathematical statements

( 2x )^2 + 4 ( 2x ) + 7 = 2 x^2 + 8 x +7

Answer:

Our L.H.S. is
(2x)^2 + 4(2x) + 7 = 4x^2 + 8x + 7
R.H.S. = 2x^2+8x+7
It is clear from the above that L.H.S. is not equal to R.H.S.
So, correct statement is
(2x)^2 + 4(2x) + 7 = 4x^2 + 8x + 7

Question:8 Find and correct the errors in the following mathematical statements

( 2 x)^2 + 5 x = 4 x +5x = 9 x

Answer:

Our L.H.S. is
\Rightarrow (2x)^{2}+5x = 4x^2+5x
R.H.S. = 9x
It is clear from the above that L.H.S. is not equal to R.H.S.
So, the correct statement is
(2x)^{2}+5x = 4x^2+5x

Question:9 Find and correct the errors in the following mathematical statements

( 3x +2 )^2 = 3 x ^2 + 6x + 4

Answer:

LHS IS

(3x + 2)^{2 } = (3x)^{2} + 2(3x)(2) +(2)^{2} using (a + b)^{2 } = (a)^{2} + 2(a)(b) +(b)^{2}
= 9x^2 + 12x + 4

RHS IS

3 x ^2 + 6x + 4

\boldsymbol{LHS} \neq \boldsymbol{RHS}

Correct statement is

(3x + 2)^{2 } = (3x)^{2} + 2(3x)(2) +(2)^{2} = 9x^2 + 12x + 4

Question:10(a) Find and correct the errors in the following mathematical statements

Substituting x = -3 in x ^ 2 + 5 x + 4 \: \: gives \: \: ( -3 ) ^ 2 + 5 ( -3 ) + 4 = 9 + 2 + 4 = 15

Answer:

We need to substitute x = -3 in

x^{2}+5x+4
=(-3)^{2}+5(-3)+4
= 9 - 15 + 4
= -2 \neq 15

so the given statement is wrong
Correct statement is (-3)^{2}+5(-3)+4= -2

Question:10(b) find and correct the errors in the following mathematical statements

Substituiting x = -3 in x ^2 -5 x + 4 \: \: gives \: \: ( -3)^2 - 5 ( -3 ) + 4 = 9 - 15 + 4 = -2

Answer:

We need to substitute x = -3 in x^2 - 5x + 4
= (-3)^2-5(-3) + 4
= 9 + 15 + 4=28
so the given statement is wrong
Correct statement is

x^2 - 5x + 4=28

Question:10(c) find and correct the errors in the following mathematical statements

Substituting x = - 3 in x ^2 + 5 x \: \: gives \: \: ( -3 ) ^ 2 + 5 ( -3 ) = -9-15 = -24

Answer:

We need to Substitute x = - 3 in x^{2} + 5x
= (-3)^{2} + 5(-3)
= 9 - 15
= - 6 \neq R.H.S
Correct statement is Substitute x = - 3 in x^{2} + 5x gives -6

Question:11 Find and correct the errors in the following mathematical statements

( y - 3 ) ^ 2 = y ^ 2 -9

Answer:

Our L.H.S. is (y - 3 )^{2}
= (y )^{2} + 2(y)(-3) + (-3)^{2} using (a-b)^{2} = (a )^{2} + 2(a)(-b) + (-b)^{2}
= y^{2} - 6x + 9 \neq R.H.S.

Correct statement is

(y - 3 )^{2} = y^{2} - 6x + 9

Question:12 Find and correct the errors in the following mathematical statements

( z+5 ) ^2 = z^2 + 25

Answer:

Our L.H.S. is (z+5)^{2}
= (z)^{2} + 2(z)(5) + (5)^{2} using (a+b)^{2} = (a)^{2} + 2(a)(b) + (b)^{2}
= (z)^{2} + 10z + 25 \neq R.H.S.
Correct statement is

(z+5)^{2} = (z)^{2} + 10z + 25

Question:13 Find and correct the errors in the following mathematical statements.

( 2a +3b ) ( a-b) = 2 a ^2 - 3 b^2

Answer:

Our L.H.S. is (2a + 3b)(a -b)
= 2a^{2} -2ab + 3ab - 3b^{2}
= 2a^{2} +ab - 3b^{2} \neq R.H.S.
Correct statement is (2a + 3b)(a -b) = 2a^{2} +ab - 3b^{2}

Question:14 Find and correct the errors in the following mathematical statements.

( a + 4 ) ( a +2 ) = a ^ 2 + 8

Answer:

Oue L.H.S. is (a + 4)(a + 2)
= a^{2} + 2a + 4a + 8
= a^{2} + 6a + 8 \neq R.H.S.
Correct statement is (a + 4)(a + 2) = a^{2} + 6a + 8

Question:15 Find and correct the errors in the following mathematical statements.

(a - 4 ) ( a - 2 )= a ^2 - 8

Answer:

Our L.H.S. is (a - 2) (a - 4)
= a^{2} - 4a - 2a + 8
= a^{2} - 6a+ 8 \neq R.H.S.
Correct statement is (a - 2) (a - 4) = a^{2} - 6a+ 8

Question:16 Find and correct the errors in the following mathematical statements.

\frac{3 x ^2}{3 x ^2 } = 0

Answer:

Our L.H.S. is
\Rightarrow \frac{3x^{2}}{3x^{2}}
R.H.S. = 0
It is clear from the above that L.H.S. is not equal to R.H.S.
So, correct statement is
\frac{3x^{2}}{3x^{2}} = 1

Question:17 Find and correct the errors in the following mathematical statements.

\frac{3 x ^2 + 1 }{3 x ^2 } = 1+1 = 2

Answer:

Our L.H.S. is
\Rightarrow \frac{3x^2+1}{3x^2}
R.H.S. = 2
It is clear from the above stattement that L.H.S. is not equal to R.H.S.
So, correct statement is
\frac{3x^{2}+1}{3x^{2}} = 1 + \frac{1}{3x^{2}} = \frac{3x^{2}+1}{3x^{2}}

Question:18 find and correct the errors in the following mathematical statements.

\frac{3 x }{3 x +2 } = 1/2

Answer:

Our L.H.S.

\Rightarrow \frac{3x}{3x+2}

R.H.S. = 1/2

It can be clearly observed that L.H.S is not equal to R.H.S

So, the correct statement is,

\frac{3x}{3x+2} = \frac{3x}{3x+2}

Question:19 find and correct the errors in the following mathematical statements

\frac{3}{4x +3}= \frac{1}{4x }

Answer:

Our L.H.S. is \Rightarrow \frac{3}{4x+3} = \frac{3}{4x+3} \neq R.H.S.

Correct statement is \frac{3}{4x+3} = \frac{3}{4x+3}

Question:20 find and correct the errors in the following mathematical statements

\frac{4 x + 5 }{4x } = 5

Answer:

Our L.H.S. is \Rightarrow \frac{4x+5}{4x} = \frac{4x}{4x} + \frac{5}{4x} = 1 + \frac{5}{4x} \neq R.H.S.

Correct statement is \frac{4x+5}{4x} = 1 + \frac{5}{4x} = \frac{4x+5}{4x}

Question:21 find and correct the errors in the following mathematical statements

\frac{7x +5}{5} = 7x

Answer:

Our L.H.S. is \Rightarrow \frac{7x+5}{5} = \frac{7x}{5} + \frac{5}{5} = \frac{7x}{5} + 1 \neq R.H.S.

Correct statement is \frac{7x+5}{5} = \frac{7x}{5} + 1 = \frac{7x+5}{5}

Factorization class 8 NCERT solutions - Topics

  • What is Factorization?
  • Division of Algebraic Expressions
  • Division of Algebraic Expressions Continued(Polynomial divide; Polynomial)
  • Can you Find the Error?

NCERT Solutions for Class 8 Maths - Chapter Wise

Key Features of Factorization Class 8 Solutions

Comprehensive Coverage: Maths chapter 14 class 8 solutions cover all topics and concepts related to factorization as per the Class 8 syllabus.

Step-by-Step Solutions: Class 8 maths ch 14 question answer are detailed, step-by-step explanations for each problem, making it easy for students to understand and apply mathematical concepts related to factorization.

Variety of Problems: A wide range of problems, including exercises and additional questions, to help students practice and test their understanding of factorization methods are discussed in ch 14 maths class 8.

NCERT Solutions for Class 8 - Subject Wise

Factorization is a key skill to solve a problem where you need to find the value of x. It will strengthen your foundations of algebra, trigonometry, calculus, and higher class maths. It has a lot of applications like calculation, make multiplication easy, prime factorization, finding LCM and HCF, solving polynomial equations, quadratic equations, and simplifying expression, etc. In NCERT solutions for Class 8 Maths chapter 14 Factorizations, you will come across some applications like simplifying expressions and solving quadratic equations. Some important expressions from NCERT solutions for Class 8 Maths chapter 14 Factorizations are given below which you should remember.

  • a^{2}+2 a b+b^{2}=(a+b)^{2}
  • a^{2}-2 a b+b^{2}=(a-b)^{2}
  • a^{2}-b^{2}=(a+b)(a-b)
  • x^{2}+(a+b) x+a b=(x+a)(x+b)

NCERT Books and NCERT Syllabus

Frequently Asked Questions (FAQs)

1. What are the important topics of Factorization ?

Factorization of algebraic expression, division of a monomial by another monomial, division of a polynomial by a monomial, and division of a polynomial by polynomial are covered in this chapter.

2. How many chapters are there in the CBSE class 8 maths ?

There are 16 chapters starting from rational number to playing with numbers in the CBSE class 8 maths.

3. Does CBSE provide NCERT solution for class 8 ?

No, CBSE doesn't provide NCERT solutions for any class and subject.

4. Where can I find the complete solutions of NCERT for class 8 ?

Here you will get the detailed NCERT solutions for class 8 by clicking on the link.

5. Where can I find the complete solutions of NCERT for class 8 maths ?

Here you will get the detailed NCERT solutions for class 8 maths by clicking on the link.

6. Which is the official website of NCERT ?

NCERT official is the official website of the NCERT where you can get NCERT textbooks and syllabus from class 1 to 12.

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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