Find Lowest Common Multiple (LCM) of the expressions given below

Question 1
2 (x + 1) and 3 (x + 1) .
solution
We first factor the given expressions completely:
2 (x + 1) = 2 (x + 1)
3 (x + 1) = 3 (x + 1)
The LCM is made by multiplying all factors included in the factoring of the given expressions. Common factors are used once only and the one with the highest power is used.
2 is a factor in the first expressions is therefore used. x + 1 is a factor in the first and second expressions is used once only. 3 is a factor in the second expression only and is therefore used . Hence
LCM ( 2 (x + 1) , 3 (x + 1) ) = 2 · 3 (x + 1)

Question 2
2 (x  1)^{ 2} and 5 (x  1) .
solution
Factor the given expressions completely:
2 (x  1)^{ 2} = 2 (x  1)^{ 2}
5 (x  1) = 5 (x  1)
We now make the LCM by multiplying all factors included in the factoring of the given expressions. Common factors are used once only and the one with the highest power is used.
2 is a factor in the first expressions is therefore used. x  1 is a factor in the first and second expressions and the factor with the highest power which is (x  1)^{ 2} in the first expression is used. 5 is a factor in the second expression only and is therefore used. Hence
LCM ( 2 (x  1)^{ 2} , 5 (x  1) ) = 2 · 5 (x  1)^{ 2}

Question 3
x^{ 2} + 5 x + 6 and 2 x^{ 2} + 2 x  4 .
solution
Factor the given expressions completely:
x^{ 2} + 5 x + 6 = (x + 3)(x + 2)
2 x^{ 2} + 2 x  4 = 2(x  1)(x + 2)
The LCM is made by multiplying all factors included in the factoring of the given expressions. Common factors are used once only and the one with the highest power is used.
x + 3 is a factor in the first expressions is therefore used. x + 2 is a factor in the first and second expressions and therefore used once. 2 is a factor in the second expression only and is therefore used. x  1 is a factor in the second expression and is therefore used. Hence
LCM ( x^{ 2} + 5 x + 6 , 2 x^{ 2} + 2 x  4 ) = 2 · (x + 3)(x + 2)(x  1)

Question 4
3 x^{ 3}  2 x ^{ 2}  x and x  1 .
solution
Factor the given expressions completely:
3 x^{ 3}  2 x ^{ 2}  x = x (3 x + 1)(x  1)
x  1 = x  1
The LCM is by multiplying all factors included in the factoring of the given expressions. Common factors are used once only and the one with the highest power is used.
x is a factor in the first expressions is therefore used. 3 x + 1 is a factor in the first expression and is therefore used. x  1 is a factor in the first and second expression and is therefore used once only. Hence
LCM ( 3 x^{ 3}  2 x ^{ 2}  x , x  1 ) = x (3x + 1)(x  1)

Question 5
3 x^{ 3}  2 x ^{ 2}  x , 2 x ^{ 2}  2 and (x  1)^{ 2}.
solution
Factor the given expressions completely:
3 x^{ 3}  2 x ^{ 2}  x = x (3 x + 1)(x  1)
2 x ^{ 2}  2 = 2(x  1)(x + 1)
(x  1)^{ 2} = (x  1)^{ 2}
We now make the LCM by multiplying all factors included in the factoring of the given expressions. Common factors are used once only and the one with the highest power is used.
x is a factor in the first expressions is therefore used. 3 x + 1 is a factor in the first expression and is therefore used. x  1 is a factor in all three expressions and the one with the highest power which is (x  1)^{ 2} in the third expression is used. 2 is a factor in the second expression and is therefore used. x + 1 is a factor in the second expression and is therefore used. Hence
LCM ( 3 x^{ 3}  2 x ^{ 2}  x , 2 x ^{ 2}  2 , (x  1)^{ 2} ) = 2 x (3x + 1)(x  1)^{ 2}(x + 1)
