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Page No: 220

Exercise 14.1

1. Find the common factors of the given terms.
(i) 12x, 36
(ii) 2y, 22xy
(iii) 14pq, 28p2q2
(iv) 2x, 3x2, 4Â
(v) 6abc, 24ab2, 12a2bÂ
(vi) 16x3, -4x2, 32xÂ
(vii) 10 pq, 20qr, 30rp
(viii) 3x2y3, 10x3y2, 6x2y2zÂ

(i) 12x = 2Ã—2Ã—3Ã—x
36 = 2Ã—2Ã—3Ã—3
Hence, the common factors are 2, 2 and 3 = 2Ã—2Ã—3 = 12

(ii) 2y = 2Ã—y
22xy = 2Ã—11Ã—xÃ—y
Hence, the common factors are 2 and yÂ = 2Ã—y = 2y

(iii)Â 14pq = 2Ã—7Ã—pÃ—q
28p2q2Â = 2Ã—2Ã—7Ã—pÃ—pÃ—qÃ—q
Hence, the common factors are 2Ã—7Ã—pÃ—q = 14pq

(iv) 2x = 2Ã—xÃ—1
3x2Â = 3Ã—xÃ—xÃ—1
4 = 2Ã—2Ã—1
Hence, the common factor is 1.

(v) 6abc = 2Ã—3Ã—aÃ—bÃ—c
24ab2Â = 2Ã—2Ã—2Ã—3Ã—aÃ—bÃ—b
12a2b = 2Ã—2Ã—3Ã—aÃ—aÃ—b
Hence, the common factors are 2Ã—3Ã—aÃ—b = 6ab

(vi) 16x3Â = 2Ã—2Ã—2Ã—xÃ—xÃ—x
-4x2Â = (-1)Ã—2Ã—2Ã—xÃ—x
32x = 2Ã—2Ã—2Ã—2Ã—2Ã—x
Hence the common factors are 2Ã—2Ã—x = 4x

(vii) 10pq = 2Ã—5Ã—pÃ—q
20qr = 2Ã—2Ã—5Ã—qÃ—r
30rp = 2Ã—3Ã—5Ã—rÃ—p
Hence the common factors are 2Ã—5 = 10

(viii) 3x2y3Â = 3Ã—xÃ—xÃ—yÃ—yÃ—y
10x3y2 = 2Ã—5Ã—xÃ—xÃ—xÃ—yÃ—y
6x2y2z = 2Ã—3Ã—xÃ—xÃ—yÃ—yÃ—z
Hence the common factors are xÃ—xÃ—yÃ—y = x2y2

2. Factorize the following expressions.
(i) 7x - 42
(ii) 6p - 12q
(iii) 7a2Â + 14aÂ
(iv) -16z + 20z3Â
(v)20l2m + 30almÂ
(vi) 5x2y - 15xy2Â
(vii) 10a2Â - 15b2Â + 20c2Â
(viii) -4a2Â + 4ab - 4caÂ
(ix) x2yz + xy2z + xyz2Â
(x) ax2y + bxy2Â + cxyzÂ

(i) 7x - 42 = 7Ã—x - 2Ã—3Ã—7
Taking common factors from each term,
= 7(x - 2Ã—3)
= 7(x - 6)

(ii) 6p - 12q = 2Ã—3Ã—p - 2Ã—2Ã—3Ã—q
Taking common factors from each term,
= 2Ã—3(p - 2q)
= 6(p - 2q)

(iii) 7a2Â + 14a = 7Ã—aÃ—a + 2Ã—7Ã—a
Taking common factors from each term,
= 7Ã—a(a + 2)
= 7a(a + 2)

(iv) -16z + 20z3
= (-1)Ã—2Ã—2Ã—2Ã—2Ã—z + 2Ã—2Ã—5Ã—zÃ—zÃ— z
Taking common factors from each term,
= 2Ã—2Ã—z (-2Ã—2 + 5Ã—zÃ—z)
= 4z (-4 + 5z2)

(v) 20l2m + 30alm
= 2Ã—2Ã—5Ã—lÃ—lÃ—m + 2Ã—3Ã—5Ã—aÃ—lÃ—m
Taking common factors from each term,
= 2Ã—5Ã—lÃ—m(2Ã—l + 3Ã—a)
= 10 lm(2l +3a)

(vi) 5x2y - 15xy2
= 5Ã—xÃ—xÃ—y - 3Ã—5Ã—xÃ—yÃ—y (Taking common factors from each term)
= 5Ã—xÃ—y(x - 3y)
= 5xy(x - 3y)

(vii) 10a2Â - 15b2Â + 20c2
= 2Ã—5Ã—aÃ—a - 3Ã—5Ã—bÃ—b + 2Ã—2Ã—5Ã—cÃ—c
Taking common factors from each term,
= 5(2Ã—aÃ—a - 3Ã—bÃ—b + 2Ã—2Ã—cÃ—c)
= 5(2a2Â - 3b2Â + 4c2)

(viii) -4a2Â + 4ab - 4ca
= (-1)Ã—2Ã—2Ã—aÃ—a + 2Ã—2Ã—aÃ—b - 2Ã—2Ã—cÃ—a
Taking common factors from each term,
= 2Ã—2Ã—a(-a + b -c)
= 4a (-a + b - c)

(ix) x2yz + xy2z + xyz2
= xÃ—xÃ—yÃ—z + xÃ—yÃ—yÃ—z + zÃ—yÃ—zÃ—z
Taking common factors from each term,
= xÃ—yÃ—z( x + y + z)
= xyz(x + y +z)

(x) ax2y + bxy2Â + cxyz
= aÃ—xÃ—xÃ—y + bÃ—xÃ—yÃ—y + cÃ—xÃ—yÃ—z
Taking common factors from each term,
= xÃ—y(aÃ—x + bÃ—y + cÃ—z)
= xy(ax + by +cz)

3.Â Factorize:
(i) x2Â + xy + 8x + 8yÂ
(ii) 15xy - 6x + 5y -2Â
(iii) ax + bx - ay - byÂ
(iv) 15pq + 15 + 9q + 25pÂ
(v) z - 7 + 7xy -xyzÂ

(i)Â x2Â + xy + 8x + 8y
= x(x + y) + 8(x + y)
= (x + y)(x + 8)

(ii) 15xy - 6x + 5y - 2
= 3x(5y - 2) + 1(5y - 2)
= (5y -2)(3x + 1)

(iii) ax + bx - ay - by
= (ax + bx) - (ay + by)
= x(a + b) - y(a + b)
= (a + b)(x - y)

(iv) 15pq + 15 + 9q + 25p
= 15pq + 25p + 9q + 15
= 5p(3q + 5) + 3(3q + 5)
= (3q + 5)(5p + 3)

(v) z -7 + 7xy - xyz = 7xy - 7 - xyz + z
= 7(xy - 1) - z(xy - 1)
= (xy -1)(7 - z) = (-1)(1 - xy)(-1)(z - 7)
= (1 - xy)(z - 7)

Page No. 223

Exercise 14.2

1. Factorize the following expressions:(i) a2Â + 8a + 16Â
(ii) p2Â - 10p + 25Â
(iii) 25m2Â + 30m + 9Â
(iv) 49y2Â + 84yz + 36z2
(v) 4x2Â - 8x + 4
(vi) 121b2Â - 88bc + 16c2Â
(vii) (l + m)2Â - 4lmÂ [Hint: Expand (l + m)2 first]
(viii) a4Â + 2a2b2Â + b4Â

(i) a2Â + 8a + 16 = a2Â + (4 + 4)a + 4Â Ã— 4
Using identity x2Â + (a + b)x + ab = (x + a)(x + b),
Here x = a, a = 4Â and b = 4
a2Â + 8a + 16 = (a + 4)(a + 4) = (a + 4)2

(ii) p2Â - 10p + 25 = p2Â +(-5-5)p + (-5)(-5)
Using identity x2Â + (a +b)x + ab = ( x + a)(x + b),
Here x = p, a = -5 and b = -5
p2Â - 10p + 25 = (p -5)(p- 5) = (p - 5)2

(iii) 25m2Â + 30m + 9 = (5m)2Â + 2Â Ã— 5mÂ Ã— 3 + (3)2
Using identity a2Â + 2ab + b2Â = (a + b)2Â ,Â here a= 5m, b = 3
Â 25m2Â + 30m + 9 = (5m + 3)2

(iv) 49y2Â + 84yz + 36z2Â = (7y)2Â + 2Â Ã— 7yÂ Ã— 6z + (6z)2
Using identity a2Â + 2ab + b2Â = (a + b)2Â , here a = 7y, b = 6z
49y2Â + 84yz + 36z2Â = (7y + 6z)2

(v) 4x2Â - 8x + 4 = (2x)2Â - 2Â Ã— 2xÂ Ã—2 + (2)2
Using identity a2Â - 2ab + b2Â = (a - b)2Â , here a = 2x, b = 2
4x2Â - 8x + 4 = (2x - 2)2
= (2)2Â (x - 1)2Â = 4( x - 1)2

(vi)Â 121b2Â - 88bc + 16c2Â = (11b)2Â - 2Â Ã— 11bÂ Ã— 4c + (4c)2
Using identity a2Â - 2ab + b2Â = (a - b)2Â , here a = 11b, b = 4c
121b2Â - 88bc + 16c2Â = (11b - 4c)2

(vii) (l + m)2Â - 4lm
= l2Â + 2Â Ã— lÂ Ã—m + m2Â - 4lmÂ [Â âˆµ (a + b)2Â = a2Â + 2ab + b2Â ]
= l2Â + 2lm + m2Â - 4lm
= l2Â - 2lm + m2
= (l - m)2 [Â âˆµ (a- b)2Â = a2Â - 2ab + b2Â ]

(viii) a4Â + 2a2b2Â + b4Â = (a2)2Â + 2Â Ã— a2Â Ã— b2Â + (b2)2
= (a2Â + b2)2Â [âˆµ (a + b)2Â = a2Â + 2ab + bÂ ]

2. Factorize:
(i) 4p2Â - 9q2Â
(ii) 63a2Â - 112b2Â
(iii) 49x2Â - 36
(iv) 16x5Â - 144x2Â
(v) (l + m)2Â - (l -m)2Â
(vi) 9x2y2Â - 16Â
(vii) (x2Â - 2xy + y2) - z2Â
(viii) 25a2Â - 4b2Â + 28bc - 49c2Â

(i) 4p2Â - 9q2Â = (2p)2Â - (3q)2
= (2p -3q)(2p + 3q) [âˆµ a2Â - b2Â = (a - b)(a +b)]

(ii) 63a2Â - 112b2Â = 7(9a2Â - 16b2)
= 7 [ (3a)2Â - (4b)2]
= 7(3a - 4b)(3a + 4b)Â  [âˆµÂ a2Â - b2Â = (a - b)(a +b)]

(iii)Â 49x2Â - 36 = (7x)2Â - (6)2
= (7x - 6)(7x + 6)Â  [âˆµ a2Â - b2Â = (a - b)(a +b)]

(iv) 16x5Â - 144x3Â = 16x3(x2Â - 9)
= 16x3Â [(x)2Â - (3)2]
= 16x3Â (x - 3)(x + 3) [âˆµ a2Â - b2Â = (a - b)(a +b)]

(v) (l + m)2Â - (l - m)2
= [(l + m) + ( l - m)][(l + m)- (l - m)] [âˆµÂ a2Â - b2Â = (a - b)(a +b)]
= (l + m + l - m)(l + m - l +m)
= (2l) (2m) = 4lm

(vi)Â 9x2y2Â - 16 = (3xy)2Â - (4)2
= (3xy - 4)(3xy + 4) [Â âˆµ a2Â - b2Â = (a - b)(a +b)]

(vii) ( x2Â - 2xy + y2) - z2Â = ( x - y)2Â - z2Â  Â [âˆµÂ (a -b)2Â = a2Â -2ab + b2]
Â = ( x - y - z)( x - y + z) [Â âˆµ a2Â - b2Â = (a - b)(a +b)]
(viii) 25a2Â - 4b2Â + 28bs - 49c2
= 25a2Â - (4b2Â - 28bc + 49c2)
= 25a2Â - [ (2b)2Â - 2Â Ã— 2bÂ Ã— 7c + (7c)2]
= 25a2Â - (2b - 7c)2 [Â âˆµÂ (a -b)2Â = a2Â -2ab + b2]
= (5a)2Â - (2b - 7c)2
= [5a - (2b - 7c)][5a + (2b - 7c)] [Â âˆµ a2Â - b2Â = (a - b)(a +b)]
= (5a - 2b + 7c)(5a + 2b - 7c)

3. Factorize the expressions:
(i) ax2Â + bx(ii) 7p2Â + 21q2Â
(iii) 2x3Â + 2xy2Â + 2xz2
(iv) am2Â + bm2Â + bn2Â + an2Â
(v) (lm + l ) + m + 1
(vi) y( y + z) + 9 ( y + z)Â
(vii) 5y2Â - 20y - 8z + 2yz
(viii) 10ab + 4a + 5b + 2Â
(ix) 6xy - 4y + 6 - 9x

(i) ax2Â + bx = x(ax + b)

(ii) 7p2Â + 21q2Â = 7(p2Â + 3q2)

(iii) 2x3Â + 2xy2Â + 2xz2Â = 2x( x2Â + y2Â + z2)

(iv) am2Â + bm2Â + bn2Â + an2
= m2( a + b) + n2(a + b)
= (a + b )(m2Â + n2)

(v) (lm + l) + m + 1
= l(m + 1) + 1(m + 1)
= (m + 1)( l + 1)

(vi) y(y + z) + 9(y + z)
= (y + z)(y + 9)

(vii) 5y2Â - 20y - 8z + 2yz
= 5y2Â - 20y + 2yz - 8z
= 5y(y - 4) + 2z(y - 4)
= (y - 4)(5y + 2z)

(viii) 10ab + 4a + 5b + 2
= 2a(5b + 2) + 1 (5b + 2)
= (5b + 2)(2a + 1)

(ix) 6xy - 4y + 6 - 9x
= 6xy - 9x - 4y + 6
= 3x(2y - 3) - 2(2y - 3)
= (2y - 3) (3x - 2)

4. Factorize:
(i) a4Â - b4
(ii) p4Â - 81Â
(iii) x4Â - (y + z)4
(iv)x4Â - (x -z)4Â
(v) a4Â - 2a2b2Â + b4Â

(i) a4Â - b4Â = (a2)2Â - (b2)2
= (a2Â - b2)( a2Â + b2) [Â âˆµ a2Â - b2Â = (a - b)(a +b)]
= (a - b)(a + b)(a2Â + b2) [Â âˆµ a2Â - b2Â = (a - b)(a +b)]

(ii) p4Â - 81 = (p2)2Â - (9)2
= (p2Â - 9)(p2Â + 9) [âˆµÂ a2Â - b2Â = (a - b)(a +b)]
= (p2Â - 32)(p2Â + 9)
= ( p - 3)(p + 3)(p2Â + 9) [âˆµÂ a2Â - b2Â = (a - b)(a +b)]

(iii) x4Â - (y + z)4Â = (x2)2Â - [(y + z)2]2
= [x2Â - (y + z)2][ x2Â + (y + z)2] [âˆµÂ a2Â - b2Â = (a - b)(a +b)]
= [x -(y +z)][x + (y + z)][x2Â + (y + z)2] [âˆµÂ a2Â - b2Â = (a - b)(a +b)]
= (x - y - z) (x + y + z) [x2Â + (y + z)2]

(iv) x4Â - (x - z)4Â = (x2)2Â - [(x - z)2]2
= [x2Â -(x - z)2][x2Â + (x - z)2] [Â âˆµ a2Â - b2Â = (a - b)(a +b)]
= [x - (x - z)][x + (x - z)] [x2Â + (x - z)2] [âˆµÂ a2Â - b2Â = (a - b)(a +b)]
= [x - x + z] [x + x - z] [x2Â + x2Â - 2xz + z2] [âˆµÂ (a -b)2Â = a2Â -2ab + b2]
= z(2x - z) (2x2Â - 2xz + z2)

(v) a4Â - 2a2b2Â + b4Â = (a2)2Â - 2a2b2Â + (b2)2
= (a2Â - b2)2 [âˆµÂ a2Â - b2Â = (a - b)(a +b)]
= [(a - b)(a + b)]2 [âˆµÂ a2Â - b2Â = (a - b)(a +b)]
= (a -b)2Â (a + b)2Â [ (xy)mÂ = xmym]

5.Â Factorize the following expressions:
(i) p2Â + 6p + 8
(ii) q2Â - 10q + 21Â
(iii) p2Â + 6p - 16Â

(i) p2Â + 6p + 8 = p2Â + ( 4 + 2)p + 4Â Ã— 2
= p2Â + 4p + 2p + 4Â Ã—2
= p(p + 4) + 2 ( p + 4)
= (p + 4)(p + 2)

(ii)Â q2Â - 10q + 21 = q2Â - ( 7 + 3)q + 7Â Ã— 3
= q2Â - 7q - 3q + 7Â Ã— 3
= q(q - 7) - 3(q - 7)
= (q - 7)( q - 3)

(iii) p2Â + 6p - 16
= p2Â + (8 - 2)p - 8Ã—2
= p2Â + 8p - 2p - 8Ã—2
= p(p + 8) - 2(p + 8)
= ( p + 8)(p -2)

Page No. 227

Exercise 14.3

1. Carry out the following divisions:
(i) 2x4Â Ã· 56x
(ii) -36y3Â Ã· 9y2
(iii) 66pq2r3Â Ã· 11 qr2
(iv) 34x3y3x3Â Ã· 51xy2z3
(v) 12a8b8Â Ã· (-6a6b4)

Â (i)Â 2x4Â Ã· 56x
= 28x4/56x
= 28/56Â Ã—Â x4/x
= 1/2Â x3 [xmÂ Ã·Â xnÂ = xm-n]

(ii) -36y3Â Ã·Â 9y2Â = -36y3/9y2
= -36/9Â Ã— y3/y2
= -4y [xmÂ Ã·Â xnÂ = xm-n]

(iii) 66pq2r3Â Ã· 11qr2
= 66pq2r3/11qr2
= 66/11Â Ã— pq2r3/qr2
= 6pqr [xmÂ Ã·Â xnÂ = xm-n]

(iv) 34x3y3z3Â Ã· 51xy2z3
= 34x3y3z3/51xy2z3
= 34/51Â Ã—x3y3z3/xy2z3
= 2/3x2y [xmÂ Ã·Â xnÂ = xm-n]

(v) 12a8b8Â Ã· (- 6a6b4)
= 12a8b8/- 6a6b4
= 12/-6Â Ã— a8b8/a6b4
= -2a2b4 [xmÂ Ã·Â xnÂ = xm-n]

2. Divide the given polynomial by the given monomial:
(i) (5x2Â - 6x)Â Ã· 3x
(ii) (3y8Â - 4y6Â + 5y4)Â Ã· y4
(iii) 8(x3y2z2Â + x2y3z2Â + x2y2z3)Â Ã· 4x2y2z2
(iv) (x3Â + 2x2Â + 3x)Â Ã·2x
(v) (p3q6Â - p6q3)Â Ã· p3q3

(i) (5x2Â - 6x)Â Ã·3x
= (5x2Â - 6x)/3x
= 5x2/3x - 6x/3x = (5/3)x - 2 = 1/3 (5x - 6)

(ii) (3y8Â - 4y6Â + 5y4)Â Ã· y4
= (3y8Â - 4y6Â + 5y4)/ y4
= 3y8/y4Â - 4y6/y4Â + 5y4/y4Â = 3y4Â - 4y2Â + 5

(iii) 8(x3y2z2Â + x2y3z2Â + x2y2z3)Â Ã· 4x2y2z2
= {8(x3y2z2Â + x2y3z2Â + x2y2z3)}/4 x2y2z2
= 8 x3y2z2/4 x2y2z2 + 8 x2y3z2/4x2y2z2 + 8 x2y2z3/4x2y2z2
= 2x + 2y + 2z
= 2(x + y + z)

(iv)Â (x3Â + 2x2Â + 3x)Â Ã· 2x
= (x3Â + 2x2Â + 3x)/2x
= x3/2x + 2x2/2x + 3x/2x = x2/2 + 2x/2 + 3/2
= 1/2( x2Â + 2x + 3)
(v) (p3q6Â - p6q3)Â Ã· p3q3
= (p3q6Â - p6q3)/p3q3
= p3q6/p3q3Â - p6q3/p3q3Â = q3Â - p3

3. Work out the following divisions:
(i) (10x - 25)Â Ã· 5
(ii) (10x - 25)Â Ã· (2x - 5)
(iii) 10y (6y + 21)Â Ã· 5(2y + 7)
(iv) 9x2y2(3z - 24)Â Ã· 27xy(z - 8)
(v) 96abc(3a - 12)(5b - 30)Â Ã· 144(a -4)(b - 6)

(i) (10x - 25)Â Ã· 5
= (10x - 25)/5
= {5(2x - 5)}/5
= 2x -5

(ii) (10x - 25)Â Ã· (2x - 5)
= (10x - 25)/(2x - 5)
= {5(2x - 5)/(2x - 5)
= 5

(iii) 10y(6y + 21)Â Ã· 5(2y + 7)
= {10y(6y + 21)}/5(2y + 7)
= {2Ã—5Ã—yÃ— 3(2y + 7)}/5(2y + 7)
= 2Ã—yÃ—3
= 6y

(iv) 9x2y2(3z - 24)Â Ã· 27xy(z - 8)
= {9x2y2(3z - 24)}/27xy(z - 8)
= 9/27Â Ã— {xyÂ Ã— xyÂ Ã— 3(z - 8)}/xy(z - 8)
= xy

(v) 96abc(3a - 12)(5b - 30)Â Ã· 144(a- 4)(b - 6)
= {96abc(3a - 12)(5b - 30)}/144(a - 4)(b - 6)
= {12Ã—4Ã—2Ã—abcÃ— 3(a-4)Â Ã— 5(b-6)}/{12Ã—4Ã—3 (a - 4)(b - 6)
= 10abc

4. Divide as directed:
(i) 5(2x + 1)(3x + 5)Â Ã· (2x + 1)
(ii) 26xy(x + 5)(y - 4)Â Ã· 13x(y - 4)
(iii) 52pqr(p + q)(q + r)(r + p)Â Ã· 104pq(q + r)(r + p)
(iv) 20(y + 4)(y2Â + 5y + 3)Â Ã· 5(y + 4)
(v) x(x + 1)(x + 2)(x + 3)Â Ã· x(x + 1)

(i) 5(2x + 1)(3x + 5)Â Ã· (2x + 1)
= {5(2x + 1)(3x +5)}/(2x + 1)
= 5(3x + 5)

(ii) 26xy( x + 5)(y - 4)Â Ã· 13x(y - 4)
26xy( x + 5)(y -4)Â Ã· 13x(y - 4)
= {26xy(x + 5)(y - 4)}/13x(y - 4)
= {13Ã—2Ã—xy(x + 5)(y - 4)}/13x(y - 4)
= 2y(x + 5)

(iii) 52pqr( p + q)(q + r)( r + p)Â Ã· 104pq(q + r)(r + p)
= {52pqr(p + q)(q + r)( r + p)}/{52Â Ã— 2Â Ã— pq(q + r)(r + p)}
= (1/2)r (p + q)

(iv) 20( y + 4)(y2Â + 5y + 3)Â Ã· 5(y + 4)
= {20(y + 4)(y2Â + 5y + 3)}/5(y + 4)
= 4(y2Â + 5y + 3)

(v) x( x + 1)(x + 2)(x + 3)Â Ã· x(x + 1)
= {x(x + 1)(x + 2)(x + 3)}/x(x + 1)
= (x + 2)(x + 3)

5. Factorize the expressions and divide them as directed:
(i) (y2Â + 7y + 10)Â Ã· (y + 5)
(ii) (m2Â - 14m - 32)Â Ã· (m + 2)
(iii) (5p2Â - 25p + 20)Â Ã· (p - 1)
(iv) 4yz(z2Â + 6z - 16)Â Ã· 2y( z + 8)
(v) 5pq(p2Â - q2)Â Ã· 2p(p + q)
(vi) 12xy(9x2Â - 16y2)Â Ã· 4xy(3x + 4y)
(vii) 39y3(50y2Â - 98)Â Ã· 26y2(5y + 7)

(i) (y2Â + 7y + 10)Â Ã· (y + 5)
= (y2Â + 7y + 10)/(y + 5)
= {y2Â + ( 2 + 5)y + 2Â Ã— 5}/(y +5)
= (y2Â + 2y + 5y + 2Â Ã— 5)/(y + 5)
= {(y + 2)(y + 5)}/(y + 5) [âˆµ x2Â + (a+b)x + ab = (x +a)(x+b)]
= y + 2

(ii) (m2Â - 14m + 32)Â Ã· (m + 2)
= (m2Â - 14m + 32)/(m +2)
= { m2Â + (-16 + 2)m + (-16)Â Ã— 2}/(m + 2)
= {(m - 16)(m + 2)}/(m +2) [âˆµ x2Â + (a+b)x + ab = (x +a)(x+b)]
= (m - 16)

(iii) (5p2Â - 25p + 20)Â Ã· (p -1)
= (5p2Â - 25p + 20)/(p -1)
= (5p2Â - 20p -5p + 20)/(p -1)
= {5p(p - 4) -5 (p - 4)}/(p -1)
= {(5p - 5)(p - 4)}/(p -1) = {5(p -1)(p -4)}/(p - 1)
= 5 (p - 4)

(iv) 4yz (z2Â + 6z - 16)Â Ã· 2y(z + 8)
= {4yz(z2Â + 6z - 16)}/2y(z + 8)
= [4yz{z2Â + (8 - 2)z + 8Â Ã— (-2)}]/2y(z + 8)
= {4yz(z - 2)(z + 8)}/2y(z + 8) [âˆµ x2Â + (a+b)x + ab = (x +a)(x+b)]
= 2z ( z -2)

(v) 5pq(p2Â - q2)Â Ã· 2p( p + q)
= {5pq(p2Â - q2)}/2p(p + q)
= {5pq(p - q)(p + q)}/2p( p + q) [âˆµÂ a2Â - b2Â = (a - b)(a + b)]
= (5/2)q (p - q)

(vi) 12xy(9x2Â - 16y2)Â Ã· 4xy(3x + 4y)
= {12xy (9x2Â - 16y2)}/4xy(3x + 4y)
= {12xy[(3x)2Â - (4y)2]}/4xy(3x + 4y)
= {12xy(3x - 4y)(3x + 4y)}/4xy(3x + 4y) [âˆµÂ a2Â - b2Â = (a - b)(a + b)]
= 3(3x - 4y)

(vii)Â 39y3(50y2Â - 98)Â Ã· 26y2(5y + 7)
= {39y3(50y2Â - 98)}/26y2(5y + 7)
= {39y3Â Ã— 2(25y2Â - 49)}/26y2(5y + 7)
= {39y2Â Ã— 2[(5y)2Â - (7)2]}/26y2(5y + 7)
= {39y2Â Ã— 2(5y - 7)(5y + 7)}/26y2(5y + 7) [âˆµÂ a2Â - b2Â = (a - b)(a + b)]
= 3y(5y - 7)

Page No. 228

Exercise 14.4

1. Find and correct the errors in the following mathematical statements:
4(x-5) = 4x-5

L.H.S. = 4(x-5) = 4x- 20Â â‰ R.H.S.
Hence, the correct mathematical statementÂ is 4(x-5) = 4x- 20.

2. x(3x+2) = 3x2+ 2

L.H.S. =Â x(3x+2) = 3x2+ 2Â â‰ Â R.H.S.
Hence, the correct mathematical statementÂ is x(3x+2) = 3x2+ 2

3. 2x + 3y = 5xy

L.H.S. =Â 2x + 3yÂ â‰ Â R.H.S.
Hence, the correct mathematical statementÂ is 2x+ 3y = 2x+ 3y

4. x+ 2x +3x = 5x

L.H.S. = x+ 2x + 3x = 6xÂ â‰ R.H.S.
Hence, the correct mathematical statementÂ is x+ 2x + 3x = 6x.

5. 5y + 2y+ y-7y = 0

L.H.S. = 5y + 2y+ y - 7y = 8y-7y = yÂ â‰ Â R.H.S.
Hence, the correct mathematical statementÂ is 5y+ 2y+y- 7y = 4

6. 3x+2x = 5Â x2

L.H.S. = 3x+ 2x = 5xÂ â‰ Â R.H.S.
Hence the correct mathematical statementÂ is 3x+ 2x = 5x

7. (2x)2+ 4(2x) + 7 = 2x2+ 8x+ 7

L.H.S. =Â (2x)2Â + 4(2x) + 7 = 4x2Â + 8x+ 7Â â‰ Â R.H.S.
Hence, the correct mathematical statementÂ is (2x)2Â + 4(2x) + 7 = 4x2Â + 8x+ 7

8. (2x)2+ 5x = 4x+ 5x = 9x

L.H.S. =Â (2x)2Â + 5x = 4x2+ 5xÂ â‰ Â R.H.S.
Hence the correct mathematical statementÂ is (2x)2Â + 5x = 4x2+ 5x.

9. (3x + 2)2= 3x2Â + 6x + 4

L.H.S. =Â (3x + 2)2Â = 3x2Â + 2Â Ã— 3xÂ Ã— 2+ (2)2
= 9x2Â + 12x + 4Â â‰ Â RHS
Hence, the correct mathematical statementsisÂ (3x + 2)2Â = 9x2Â + 12X + 4Â Ã— 3x

10. Substituting x = -3Â in:
(a)Â x2Â + 5X + 4Â gives (-3)2Â + 5(-3)Â + 4 = 9+ 2+4 = 15
(b)Â x2Â - 5X + 4 gives (-3)2Â - 5(-3) + 4 = 9 - 15 + 4 = -2
(c)Â x2Â + 5XÂ gives (-3)2Â + 5(-3) = -9 - 15 = -24

(a)Â L.H.S. =Â x2Â + 5x + 4
Putting x = -3Â in given expression,
Â =Â (-3)2Â + 5(-3)Â + 4 = 9 - 15 + 4 = -2 R.H.S.
Hence,Â x2Â + 5x + 4 givesÂ (-3)2Â + 5(-3)Â + 4 = 9 - 15 + 4 = -2

(b)Â L.H.S. =Â x2Â - 5X + 4
Putting x = -3 cin given expression,
Â =Â (-3)2Â - 5(-3)Â + 4 = 9 + 15 + 4 = 28 â‰ Â R.H.S.
HenceÂ x2Â â€“ 5x + 4 givesÂ (-3)2Â - 5(-3)Â + 4 = 9 + 15 + 4 = 28

(c)Â L.H.S. =Â x2Â + 5X
Putting x= -3Â in given expression,
Â =Â (-3)2Â + 5(-3)Â = 9 - 15 = -6Â â‰ Â R.H.S.
Hence,Â x2Â + 5X givesÂ (-3)2Â + 5(-3)Â = 9 - 15 = -6

11. (y-3)2= y2Â - 9Â

L.H.S. =Â (y-3)2Â = y2Â - 2Â Ã— yÂ Ã— 3 +(3)2Â Â [ (a-b)2Â = a2Â - 2ab + b2]
= y2Â - 6y + 9 â‰ Â R.H.S.
Hence, the correct statementÂ is (y-3)2Â = y2Â - 6y + 9

12. (z+5)2Â = z2Â + 25

L.H.S. =Â (z+5)2Â = z2Â +Â 2Â Ã— zÃ—5+ (5)2
= z2Â + 10z +25 [ (a-b)2Â = a2Â - 2ab + b2]
Hence, the correct statement isÂ (z+5)2Â =Â z2Â + 10z + 25

13. (2a +3b)(a-b) = 2a2- 3b2

L.H.S. = (2a + 3b)(a-b) = 2a(a-b) + 3b(a-b)
= 2a2Â - 2ab + 3ab - 3b2
= 2a2Â + ab - 3b2 â‰ Â R.H.S.
Hence, the correct statement isÂ (2a +3b)(a-b) = 2a2Â + ab - 3b2

14. (a + 4) (aÂ + 2) = a2+ 8

L.H.S. = (a+4)(a+2) =a(a+2) + 4(a+2)
= a2Â + 2a + 4a + 8 = a2Â + 6a + 8Â â‰ Â R.H.S.
Hence, the correct statement is (a+4)(a+2) = a2+6a+ 8

15. (a-4)(a-2) = a2- 8

L.H.S. = (a-4)(a-2) = a(a-2)-4(a-2)
Â = a2Â - 2a -4a+8 = a2- 6a + 8Â â‰ Â R.H.S
Hence, the correct statement is (a-4)(a-2) = a2- 6a + 8

16. 3x2/3x2= 0Â

L.H.S. =Â 3x2/3x2Â =1/1 = 1Â â‰ Â R.H.S.
Hence, the correct statement isÂ 3x2/3x2Â =1

17. 3x2Â + 1 / 3x2Â = 1+ 1 = 2

L.H.S. =Â 3x2Â + 1 / 3x2Â = 3x2/ 3x2Â + 1/3x2
= 1 + 1 / 3x2 R.H.S.
Hence, the correct statement isÂ 3x2Â + 1 / 3x2Â =Â 1 + 1/3x2Â

18. 3x/(3x+2) = 1/2

L.H.S. =Â 3x/(3x+2)Â â‰ Â R.H.S.
Hence, the correct statement isÂ 3x/(3x+2) =Â 3x/(3x+2)

19. 3/(4x+3) = 1/4x

L.H.S. =Â 3/(4x+3) â‰ Â R.H.S.
Hence, the correct statement isÂ 3/(4x+3)Â =Â 3/(4x+3)

20. (4x+5)/4x = 5

L.H.S. =Â (4x+5)/4x = 4x/4x + 5/4x = 1 + 5/4x â‰ R.H.S.
Hence the correct statement is (4x+ 5)/4x = 1 + 5/4x

21. (7x+5)/5 = 7x

L.H.S. =Â (7x+5)/5 = 7x/5 + 5/5 = 7x/5 + 1Â â‰ Â R.H.S.
Hence, the correct statement isÂ (7x+5)/5 = 7x/5 +1

NCERT Solutions for Class 8 Maths Chapter 14 Factorisation

Chapter 14 Factorisation NCERT Solutions are accurate and detailed which will increase concentration and you can solve questions of supplementary books easily. Factorisation means write an expression as a product of its factors. When we factorise an expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.

â€¢ Some expression can easily be factorised using these identities:
(i)Â  a2 + 2ab + b2 = (a + b)2
(ii) a2 â€“ 2ab + b2 = (a â€“ b)2
(iii) a2 â€“ b2 = (a â€“ b)(a + b)
(iv) x2 + (a + b)x + ab = (x + a)( x+ b)

â€¢ The number 1 is a factor of every algebraic term also, but it is shown only when needed.

Below are exercisewise Class 8 Maths NCERT Solutions by which you can understand the concepts behind the questions and easily solve them.

Studyrankers experts have prepared these NCERT Solutions with the sole intention of helping students in better manner. These NCERT Solutions for Class 8are updated as per the latest marking scheme released by CBSE.

NCERT Solutions for Class 8 Maths Chapters:

 Chapter 1 Rational Numbers Chapter 2 Linear Equations in Variable Chapter 3 Understanding Quadrilaterals Chapter 4 Practical Geometry Chapter 5 Data Handling Chapter 6 Squares and Square Roots Chapter 7 Cubes and Cube RootsÂ Chapter 8 Comparing Quantities Chapter 9 Algebraic Expressions and Identities Chapter 10 Visualising Solid Shapes Chapter 11 Mensuration Chapter 12 Exponents and Powers Chapter 13 Direct and Inverse Proportions Chapter 15 Introduction to Graphs Chapter 16 Playing with Numbers

FAQ onÂ ChapterÂ 14 Factorisation

Factorise 9x + 18y + 6xy + 27.

Here, we have a common factor 3 in all the terms.
âˆ´ 9x + 18y + 6xy + 27 = 3[3x + 6y + 2xy + 9]
We find that 3x + 6y = 3(x + 2y) and 2xy + 9 = 1(2xy + 9)
i.e. a common factor in both the groups does not eist,
Thus, 3x + 6y + 2xy + 9 cannot be factorised.
On regrouping the terms, we have
3x + 6y + 2xy + 9 = 3x + 9 + 2xy + 6y
= 3(x + 3) + 2y(x + 3)
= (x + 3)(3 + 2y)
Now, 3[3x + 6y + 2xy + 9] = 3[(x + 3)(3 + 2y)]
Thus, 9x + 18y + 6xy + 27 = 3(x + 3)(2y + 3)

Write 10y as irreducible factor form.

We haveÂ Â
10 = 2 Ã— 5
xy = x Ã— y
âˆ´Â  10xy = 2 Ã— 5 Ã— x Ã— y.

Factorise: x â€“ 9 + 9zy â€“ xyz.

By regrouping, we have
x â€“ 9 + 9zy â€“ xyz = x â€“ 9 â€“ xyz + 9zy
= 1(x â€“ 9) â€“ yz(x â€“ 9)
= (x â€“ 9)(1 â€“ yz)
= (x â€“ 9)(1 â€“ yz).

Factorise: 54x2 â€“ 96y2

We have 54x2 â€“ 96y2 = 6[9x2 â€“ 16y2]
= 6[(3x)2 â€“ (4y)2]
= 6[(3x + 4y)(3x â€“ 4y)] [Using a2 â€“ b2 = (a + b)(a â€“ b)]
Thus, 54x2 â€“ 96y2 = 6 (3x + 4y)(3x â€“4y).
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