- Find the LCM of the following:
(i) 2x + 6, x2 + 3x
Solution:
2x + 6 = 2 (x + 3)
x2 + 3x = x (x + 3)
LCM = 2x (x + 3)
(ii) x2y + xy2, x 2+ xy
Solution:
x2y + xy2 = xy (x + y)
x 2+ xy = x ( x+ y)
LCM = xy (x + y)
(iii) 3x2 – 75, 2x3 + 250
Solution:
3x2 – 75 = 3 (x2 – 25)
= 3 (x2 – 52)
= 3 (x + 5) (x – 5)
2x3 + 250 = 2 (x3 + 125)
LCM = 2 x 3 (x + 5) (x3 + 125)
= 6 (x + 5) (x3 + 125)
(iv) a2 – 1, a4 – 1, a8 – 1
Solution:
a2 – 1 = (a + 1) (a – 1)
a4 – 1 = (a2 + 1) (a2 – 1)
= (a2 + 1) (a + 1) (a – 1)
a8 – 1 = (a4 + 1) (a4 – 1)
= (a4 + 1) [(a2)2 – 12]
= (a4 + 1) (a2 + 1) (a2 – 1)
= (a4 + 1) (a2 + 1) (a + 1) (a – 1)
LCM = (a + 1) (a – 1) (a2 + 1) (a4 + 1)
(v) m2 – n2 , 3m2 – 3mn
Solution:
m2 – n2 = (m + n) (m – n)
3m2 – 3mn = 3m (m – n)
LCM = 3m (m + n) (m – n)
(vi) 5(y2 – z2), y2 + 2yz + z2
Solution:
5(y2 – z2) = 5 (y – z) (y + z)
y2 + 2yz + z2 = (y + z)2 [(a + b)2 = a2 + 2ab + b2]
LCM = 5 (y – z) (y + z)2
(vii) x3 + 8, x2 – 4
Solution:
x3 + 8 = x3 + 23
= (x + 2) (x2 – 2x +4)
x2 – 4 = x2 + 22
= (x + 2) (x – 2)
LCM = (x + 2) (x – 2) (x2 – 2x +4) = (x+2)(x3+8)
(viii) 3(a + b)2, 5(a – b )2, 2(a2 – b2)
Solution:
3(a + b)2 = 3(a + b) (a + b)
5(a – b)2 = 5(a – b ) (a – b)
2(a2 – b2) = 2(a + b) (a – b)
LCM = 2 x 3 x 5 (a + b)2 (a – b)2
LCM =30 (a + b)2 (a – b)2 = 30(a2-b2)2
(ix) 8x3 – y3, ab (4x2 + 2xy + y2), bc (4x2 – y2)
Solution:
8x3 – y3 = (2x)3 – y3
= (2x – y) [(2x)2 + 2.x.y + y2]
= (2x – y) (4x2 + 2xy + y2)
ab (4x2 + 2xy + y2)
bc (4x2 – y2) = bc [(2x)2 – y2] = bc (2x + y) (2x – y)
LCM = abc (2x – y) (2x + y) (4x2 + 2xy + y2) = abc(2x+y)(8x3-y3)
(x) 21(x – 1)2 , 35 (x4 – x2) , 14 (x4 – x)
Solution:
21(x – 1)2 = 3 x 7 (x – 1)2
35 (x4 – x2) = 5 x 7x2 (x2 – 1)
= 5 x 7x2 (x + 1) (x – 1)
14 (x4 – x) = 2 x 7x (x3 – 1)
= 2 x 7x (x3 – 1)
= 2×7x (x-1)(x2 + x +1)
LCM = 2 x 3 x 5 x 7×2 (x + 1)(x-1)2 (x2 + x +1)
= 210x2(x+1)(x-1)(x3-1) = 210x2(x2-1)(x3-1)
- Find the LCM of the following:
(i) x2 – 3x – 4, x2 +2x – 24
Solution:
x2 – 3x – 4
= x2 – 4x + x – 4
= x (x – 4) +1 (x – 4)
= (x – 4) (x + 1)
x2 +2x – 24
= x2 +6x – 4x – 24
= x (x + 6) – 4 (x + 6)
= (x + 6) (x – 4)
LCM = (x – 4) (x + 1) (x + 6)
(ii) x2 + 4x + 4, x2 + 5x + 6
x2 + 4x + 4
= x2 + 2x + 2x + 4
= x (x + 2) + 2 (x +2)
= (x +2) (x +2) =(x+2)2
x2 + 5x + 6
= x2 + 2x + 3x + 6
= x (x + 2) + 3 (x +2)
= (x + 2) (x +3)
LCM = (x + 2)2 (x +3)
(iii) – x2 – x + 6, – x2 + x + 2
– x2 – x + 6
= – (x2 + x – 6)
= (x2 + 3x – 2x – 6)
=[x(x + 3) – 2(x + 3)]
= (x – 2) (x + 3)
– x2 + x + 2
= – (x2 – x – 2)
= – x2+ 2x – x + 2]
= x(-x + 2) + 1(-x + 2)]
= (-x + 2) (x + 1) = (x+1)(2 – x)
LCM = (2 – x) (x + 1) (x + 3)
(iv) 6m² – 3m – 45, 6m² + 11m – 10
Solution:
6m2 – 3m – 45
= 3(2m2 – 3m – 15)
= 3(2m2 – 6m + 5m – 15)
= 3[(2m (m – 3) + 5 (m – 3)]
= 3(m – 3)(2m + 5)
6m2 + 11m – 10
= 6m2 + 15m – 4m – 10
= 3m (2m + 5) – 2 (2m + 5)
= (2m + 5) (3m – 2)
LCM = 3(m – 3) (2m + 5) (3m – 2)
(v) 10x3 + 6x2 – 28x , 9x3 + 15x2 – 6x
Solution:
10x3 + 6x2 – 28x
= 2x (5x2 + 3x – 14)
= 2x (5x2 + 10x – 7x – 14)
= 2x [5x (x + 2) – 7 (x + 2)]
= 2x (x + 2) (5x – 7)
9x3 + 15x2 – 6x
= 3x (3x2 + 5x – 2)
= 3x (3x2 + 6x – x – 2)
= 3x [3x (x + 2) – 1 (x + 2)]
= 3x (x + 2) (3x – 1)
LCM = 2 x 3 x x (x + 2) (3x – 1) (5x – 7)
= 6x (x + 2) (3x – 1) (5x – 7)
(vi) 6a3 + 60a2 + 150a, 3a4 + 12a3 – 15a2
6a3 + 60a2 + 150a
= 6a (a3 + 10a2 + 25)
= 6a (a3 + 2.5a2 + 52)
= 6a (a + 5)2
3a4 + 12a3 – 15a2
= 3a2 (a2 + 4a – 5)
= 3a2 (a2 + 5a – a – 5)
= 3a2 [a (a + 5) – 1 (a + 5)]
= 3a2 (a + 5) (a – 1)
LCM = 6a2 (a + 5)2 (a – 1)
(vii) 12x4 + 324x, 36x3 + 90x2 – 54x
Solution:
12x4 + 324x
= 12x (x3 + 27)
= 12x (x3 + 33)
= 12x (x + 3) (x2 – 3x + 9)
= 22 x 3x (x + 3) (x2 – 3x + 9)
36x3 + 90x2 – 54x
= 18x (2x2 + 5x – 3)
= 18x (2x2 + 6x – x – 3)
= 18x [2x (x + 3) – 1 (x + 3)
= 18x (x + 3) (2x – 1)
= 2 x 33 x (x + 3) (2x – 1)
LCM = 22 x 33 x (x + 3) (2x – 1) (x2 – 3x + 9)
= 36x (x3 – 27) (2x – 1)
(viii) a2 – 3a + 2, a3 – a2 – 4a + 4, a (a3 – 8)
Solution:
a2 – 3a + 2 = a2 – 2a – a + 2
= a (a – 2) – 1 (a – 2)
= (a – 1) (a – 2)
a3 – a2 – 4a + 4 = a2 (a – 1) – 4 (a – 1)
= (a – 1) (a2 – 4)
= (a – 1) (a2 – 22)
= (a – 1) (a – 2) (a + 2)
a (a3 – 8) = a (a3 – 23)
= a (a – 2) (a2 + 2a + 4)
LCM = a (a – 1) (a – 2) (a + 2) (a2 + 2a + 4)
= a (a – 1) (a + 2) (a3 – 8)
(ix) 4x3 + 4x2 – x – 1, 8x3 – 1, 8x2 – 2x – 1
4x3 + 4x2 – x – 1
= 4x2(x + 1) – 1 (x + 1)
= (x +1) (4x2 – 1)
= (x – 1) [(2x)2 – 1]
= (x – 1) (2x + 1) (2x – 1)
8x3 – 1 = (2x)3 – 1
= (2x – 1) [(2x)2 + 2x.1 + 12]
=(2x – 1) (4x2 + 2x + 1)
LCM = (x + 1) (2x – 1) (2x + 1) (4x2 + 2x + 1)
= (x + 1) (2x + 1) (8x3 – 1)
(x) m2 – 9m – 22, m2 – 8m – 33, m2 + 5m + 6
Solution:
m2 – 9m – 22
= m2 – 11m + 2m – 22
= m (m – 11) + 2 (m – 11)
= (m – 11) (m + 2)
m2 – 8m – 33
= m2 – 11m + 3m – 33
= m (m – 11) + 3 (m – 11)
= (m – 11) (m + 3)
m2 + 5m + 6 = m2 + 3m + 2m + 6
= m (m + 3) + 2 (m + 3)
= (m + 3) (m +2)
LCM = (m – 11) (m + 2) (m + 3)
(xi) 6 (x2 + 2xy – 3y3), 4(x2 – 3xy + 2y2), 8(x2 + xy – 6y2)
Solution:
6 (x2 + 2xy – 3y3) = 6 (x2 + 3xy – xy – 3y2)
= 6 [x (x + 3y) – y (x + 3y)]
= 6 (x + 3y) (x – y)
= 2 x 3 (x + 3y) (x – y)
= 4 (x2 – 3xy +2 y2)
4 (x2 – 3xy + 2y2) = 4 (x2 – 2xy – xy + 2y2)
= 4 [x (x – 2y) –y (x – 2y)]
= 4(x – 2y) (x – y)
= 22 (x – 2y) (x – y)
8(x2 + xy – 6y2) = 8(x2 + 3xy – 2xy – 6y2)
= 8 [x (x + 3y) – 2y (x +3y)]
= 8 (x + 3y) (x – 2y)
= 23 (x + 3y) (x – 2y)
LCM = 23 x 3 (x + 3y) (x – y) (x – 2y)
= 24 (x – y) (x – 2y) (x + 3y)
(xii) pq2 (x2 + x – 20), p2q(x2 – 3x – 4), p2q2 (x2 + 2x + 1)
pq2 (x2 + x – 20)
= pq2 (x2 + 5x – 4x – 20)
= pq2 [x (x + 5) – 4(x + 5)]
= pq2 (x + 5) (x – 4)
p2q(x2 – 3x – 4)
= p2q(x2 – 4x + x – 4)
= pq2 [x (x – 4) + 1 (x – 4)]
= pq2 (x – 4) (x + 1)
p2q2 (x2 + 2x + 1) = p2q2 (x2 + 2x + 1)
= p2q2 (x + 1)2
= p2q2 (x + 1)2 [(a+ b)2 = a2 + 2ab + b2]
LCM = p2q2 (x + 5) (x – 4) (x + 1)2
1 thought on “HCF AND LCM – EXERCISE 3.3.4 – Class 9”
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