**Find the LCM of the following:**

**(i) 2x + 6, x**^{2}**+ 3x **

Solution:

2x + 6 = 2 (x + 3)

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

**LCM = 2x (x + 3**)

**(ii) x**^{2}**y + xy**^{2}**, x **^{2}**+ xy **

Solution:

x^{2}y + xy^{2} = xy (x + y)

x ^{2}+ xy = x ( x+ y)

**LCM = xy (x + y) **

**(iii) **3x^{2} – 75, 2x^{3} + 250

Solution:

3x^{2} – 75 = 3 (x^{2} – 25)

= 3 (x^{2} – 5^{2})

= 3 (x + 5) (x – 5)

2x^{3} + 250 = 2 (x^{3} + 125)

LCM = 2 x 3 (x + 5) (x^{3} + 125)

= 6 (x + 5) (x^{3} + 125)

**(iv) a**^{2}**– 1, a**^{4}**– 1, a**^{8}**– 1 **

Solution:

a^{2} – 1 = (a + 1) (a – 1)

a^{4} – 1 = (a^{2} + 1) (a^{2} – 1)

= (a^{2} + 1) (a + 1) (a – 1)

a^{8} – 1 = (a^{4} + 1) (a^{4} – 1)

= (a^{4} + 1) [(a^{2})^{2} – 1^{2}]

= (a^{4} + 1) (a^{2} + 1) (a^{2} – 1)

= (a^{4} + 1) (a^{2} + 1) (a + 1) (a – 1)

**LCM = (a + 1) (a – 1) (a**^{2}**+ 1) (a**^{4}**+ 1) **

**(v) m**^{2}**– n**^{2}**, 3m**^{2}**– 3mn **

Solution:

m^{2} – n^{2} = (m + n) (m – n)

3m^{2} – 3mn = 3m (m – n)

**LCM = 3m (m + n) (m – n) **

**(vi) 5(y**^{2}**– z**^{2}**), y**^{2}**+ 2yz + z**^{2 }

Solution:

5(y^{2} – z^{2}) = 5 (y – z) (y + z)

y^{2 }+ 2yz + z^{2} = (y + z)^{2} [(a + b)^{2} = a^{2} + 2ab + b^{2}]

**LCM = 5 (y – z) (y + z) ^{2}**

**(vii) x**^{3}**+ 8, x**^{2}**– 4 **

Solution:

x^{3} + 8 = x^{3} + 2^{3}

= (x + 2) (x^{2} – 2x +4)

x^{2 }– 4 = x^{2} + 2^{2}

= (x + 2) (x – 2)

**LCM = (x + 2) (x – 2) (x**^{2}**– 2x +4) = (x+2)(x ^{3}+8) **

**(viii) 3(a + b)**^{2}**, 5(a – b )**^{2}**, 2(a**^{2}**– b**^{2}**) **

Solution:

3(a + b)^{2} = 3(a + b) (a + b)

5(a – b)^{2} = 5(a – b ) (a – b)

2(a^{2} – b^{2}) = 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(a ^{2}-b^{2})^{2}**

**(ix) 8x**^{3}**– y**^{3}**, ab (4x**^{2}**+ 2xy + y**^{2}**), bc (4x**^{2}**– y**^{2}**) **

Solution:

8x^{3} – y^{3} = (2x)^{3} – y^{3}

= (2x – y) [(2x)^{2 }+ 2.x.y + y^{2}]

= (2x – y) (4x^{2} + 2xy + y^{2})

ab (4x^{2 }+ 2xy + y^{2})

bc (4x^{2 }– y^{2}) = bc [(2x)^{2} – y^{2}] = bc (2x + y) (2x – y)

**LCM = abc (2x – y) (2x + y) (4x**^{2}**+ 2xy + y**^{2}**) = abc(2x+y)(8x ^{3}-y^{3})**

**(x) 21(x – 1)**^{2}**, 35 (x**^{4}**– x**^{2}**) , 14 (x**^{4}**– x) **

Solution:

21(x – 1)^{2} = 3 x 7 (x – 1)^{2 }

35 (x^{4 }– x^{2}) = 5 x 7x^{2} (x^{2} – 1)

= 5 x 7x^{2} (x + 1) (x – 1)

14 (x^{4} – x) = 2 x 7x (x^{3} – 1)

= 2 x 7x (x^{3} – 1)

= 2×7*x* (x-1)(x^{2} + x +1)

LCM = 2 x 3 x 5 x 7×2 (x + 1)(x-1)^{2} (x2 + x +1)

= 210x^{2}(x+1)(x-1)(x^{3}-1) = 210x^{2}(x^{2}-1)(x^{3}-1)

**Find the LCM of the following:**

**(i) x**^{2}**– 3x – 4, x**^{2}**+2x – 24 **

Solution:

x^{2} – 3x – 4

= x^{2} – 4x + x – 4

= x (x – 4) +1 (x – 4)

= (x – 4) (x + 1)

x^{2} +2x – 24

= x^{2 }+6x – 4x – 24

= x (x + 6) – 4 (x + 6)

= (x + 6) (x – 4)

**LCM = (x – 4) (x + 1) (x + 6)**

**(ii) x**^{2 }**+ 4x + 4, x**^{2}**+ 5x + 6 **

x^{2} + 4x + 4

= x^{2} + 2x + 2x + 4

= x (x + 2) + 2 (x +2)

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

x^{2} + 5x + 6

= x^{2} + 2x + 3x + 6

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

= (x + 2) (x +3)

**LCM = (x + 2)**^{2}**(x +3) **

**(iii) – x**^{2}**– x + 6, – x**^{2}**+ x + 2 **

– x^{2} – x + 6

= – (x^{2} + x – 6)

= (x^{2} + 3x – 2x – 6)

=[x(x + 3) – 2(x + 3)]

= (x – 2) (x + 3)

– x^{2} + x + 2

= – (x^{2} – x – 2)

= – x^{2}+ 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:

6m^{2} – 3m – 45

= 3(2m^{2} – 3m – 15)

= 3(2m^{2} – 6m + 5m – 15)

= 3[(2m (m – 3) + 5 (m – 3)]

= 3(m – 3)(2m + 5)

6m^{2} + 11m – 10

= 6m^{2} + 15m – 4m – 10

= 3m (2m + 5) – 2 (2m + 5)

= (2m + 5) (3m – 2)

**LCM = 3(m – 3) (2m + 5) (3m – 2) **

**(v) 10x**^{3}**+ 6x**^{2 }**– 28x , 9x**^{3}**+ 15x**^{2}**– 6x **

Solution:

10x^{3} + 6x^{2} – 28x

= 2x (5x^{2} + 3x – 14)

= 2x (5x^{2} + 10x – 7x – 14)

= 2x [5x (x + 2) – 7 (x + 2)]

= 2x (x + 2) (5x – 7)

9x^{3} + 15x^{2 }– 6x

= 3x (3x^{2} + 5x – 2)

= 3x (3x^{2} + 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) 6a**^{3}**+ 60a**^{2}**+ 150a, 3a**^{4}**+ 12a**^{3}**– 15a**^{2 }

6a^{3} + 60a^{2} + 150a

= 6a (a^{3} + 10a^{2} + 25)

= 6a (a^{3} + 2.5a^{2} + 5^{2})

= 6a (a + 5)^{2}

3a^{4 }+ 12a^{3 }– 15a^{2 }

= 3a^{2} (a^{2} + 4a – 5)

= 3a^{2} (a^{2} + 5a – a – 5)

= 3a^{2} [a (a + 5) – 1 (a + 5)]

= 3a^{2} (a + 5) (a – 1)

**LCM = 6a**^{2}**(a + 5)**^{2}**(a – 1) **

**(vii) 12x**^{4 }**+ 324x, 36x**^{3 }**+ 90x**^{2}**– 54x **

Solution:

12x^{4} + 324x

= 12x (x^{3} + 27)

= 12x (x^{3} + 3^{3})

= 12x (x + 3) (x^{2 }– 3x + 9)

= 2^{2} x 3x (x + 3) (x^{2} – 3x + 9)

36x^{3} + 90x^{2} – 54x

= 18x (2x^{2} + 5x – 3)

= 18x (2x^{2} + 6x – x – 3)

= 18x [2x (x + 3) – 1 (x + 3)

= 18x (x + 3) (2x – 1)

= 2 x 3^{3} x (x + 3) (2x – 1)

**LCM = 2**^{2}**x ****3**^{3}**x (x + 3) (2x – 1) (x**^{2}**– 3x + 9) **

**= 36x (x****3 ****– 27) (2x – 1)**

** **

**(viii) a**^{2}**– 3a + 2, a**^{3 }**– a**^{2 }**– 4a + 4, a (a**^{3}**– 8) **

Solution:

a^{2} – 3a + 2 = a^{2} – 2a – a + 2

= a (a – 2) – 1 (a – 2)

= (a – 1) (a – 2)

a^{3} – a^{2} – 4a + 4 = a^{2 }(a – 1) – 4 (a – 1)

= (a – 1) (a^{2} – 4)

= (a – 1) (a^{2} – 2^{2})

= (a – 1) (a – 2) (a + 2)

a (a^{3} – 8) = a (a^{3} – 2^{3})

= a (a – 2) (a^{2} + 2a + 4)

**LCM = a (a – 1) (a – 2) (a + 2) (a**^{2}**+ 2a + 4) **

**= a (a – 1) (a + 2) (a**^{3 }**– 8)**

**(ix) 4x**^{3}**+ 4x**^{2}**– x – 1, 8x**^{3 }**– 1, 8x**^{2}**– 2x – 1 **

4x^{3 }+ 4x^{2} – x – 1

= 4x^{2}(x + 1) – 1 (x + 1)

= (x +1) (4x^{2} – 1)

= (x – 1) [(2x)^{2} – 1]

= (x – 1) (2x + 1) (2x – 1)

8x^{3} – 1 = (2x)^{3} – 1

= (2x – 1) [(2x)^{2} + 2x.1 + 1^{2}]

=(2x – 1) (4x^{2} + 2x + 1)

**LCM = (x + 1) (2x – 1) (2x + 1) (4x**^{2}**+ 2x + 1) **

**= (x + 1) (2x + 1) (8x**^{3}**– 1)**

** **

**(x) m**^{2}**– 9m – 22, m**^{2}**– 8m – 33, m**^{2}**+ 5m + 6 **

Solution:

m^{2 }– 9m – 22

= m^{2} – 11m + 2m – 22

= m (m – 11) + 2 (m – 11)

= (m – 11) (m + 2)

m^{2 }– 8m – 33

= m^{2} – 11m + 3m – 33

= m (m – 11) + 3 (m – 11)

= (m – 11) (m + 3)

m^{2} + 5m + 6 = m^{2} + 3m + 2m + 6

= m (m + 3) + 2 (m + 3)

= (m + 3) (m +2)

**LCM = (m – 11) (m + 2) (m + 3) **

**(xi) 6 (x**^{2}**+ 2xy – 3y**^{3}**), 4(x**^{2}**– 3xy + 2y**^{2}**), 8(x**^{2}**+ xy – 6y**^{2}**) **

Solution:

6 (x^{2} + 2xy – 3y^{3}) = 6 (x^{2} + 3xy – xy – 3y^{2})

= 6 [x (x + 3y) – y (x + 3y)]

= 6 (x + 3y) (x – y)

= 2 x 3 (x + 3y) (x – y)

= 4 (x^{2} – 3xy +2 y^{2})

4 (x^{2} – 3xy + 2y^{2}) = 4 (x^{2} – 2xy – xy + 2y^{2})

= 4 [x (x – 2y) –y (x – 2y)]

= 4(x – 2y) (x – y)

= 2^{2} (x – 2y) (x – y)

8(x^{2} + xy – 6y^{2}) = 8(x^{2} + 3xy – 2xy – 6y^{2})

= 8 [x (x + 3y) – 2y (x +3y)]

= 8 (x + 3y) (x – 2y)

= 2^{3} (x + 3y) (x – 2y)

**LCM = 2**^{3}**x 3 (x + 3y) (x – y) (x – 2y) **

**= 24 (x – y) (x – 2y) (x + 3y)**

**(xii) pq**^{2}**(x**^{2}**+ x – 20), p**^{2}**q(x**^{2}**– 3x – 4), p**^{2}**q**^{2}**(x**^{2 }**+ 2x + 1) **

pq^{2} (x^{2} + x – 20)

= pq^{2} (x^{2} + 5x – 4x – 20)

= pq^{2} [x (x + 5) – 4(x + 5)]

= pq^{2} (x + 5) (x – 4)

p^{2}q(x^{2} – 3x – 4)

= p^{2}q(x^{2} – 4x + x – 4)

= pq^{2} [x (x – 4) + 1 (x – 4)]

= pq^{2} (x – 4) (x + 1)

p^{2}q^{2} (x^{2} + 2x + 1) = p^{2}q^{2} (x^{2 }+ 2x + 1)

= p^{2}q^{2} (x + 1)^{2}

= p^{2}q^{2} (x + 1)^{2} [(a+ b)^{2} = a^{2} + 2ab + b^{2}]

**LCM = p**^{2}**q**^{2}**(x + 5) (x – 4) (x + 1)**^{2}

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