Question

Find the LCM of $12{x^4} + 324x$ , $36{x^3} + 90{x^2} - 54x$

Answer 1

Hint: Here, we are required to find the LCM of the given two expressions. Now, in order to find the LCM, first of all, we will factorize the given expressions by taking out the common terms and using the algebraic formulas. After factoring both the expressions, we will compare them and find the required LCM by multiplying the factors of both the expressions. In case a factor is the same, we will take the highest possible power present in any of the two given expressions and find the required LCM.

Formula Used:
We will use the formula $\left( {{a^3} + {b^3}} \right) = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)$.

Complete step-by-step answer:
The first expression is: $12{x^4} + 324x$
Now, we will factorize this expression by taking out the common terms from the expression.
$ \Rightarrow 12{x^4} + 324x = 12x\left( {{x^3} + 27} \right)$
The terms inside the bracket in the RHS can also be written as:
$ \Rightarrow 12{x^4} + 324x = 12x\left( {{x^3} + {3^3}} \right)$
Now, using the formula $\left( {{a^3} + {b^3}} \right) = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)$, we get
$ \Rightarrow 12{x^4} + 324x = 12x\left( {x + 3} \right)\left( {{x^2} + {3^2} - 3x} \right)$
Here, the number 12 in the RHS can be written as a product of its factor. So,
$ \Rightarrow 12{x^4} + 324x = \left( {2 \times 2 \times 3} \right)x\left( {x + 3} \right)\left( {{x^2} - 3x + 9} \right)$
Therefore, the factors of the first expression $12{x^4} + 324x$ are:
$\left( {2 \times 2 \times 3} \right),x,\left( {x + 3} \right),\left( {{x^2} - 3x + 9} \right)$…………………………….$\left( 1 \right)$
Again, according to the question,
The second expression is: $36{x^3} + 90{x^2} - 54x$
Now, we will factorize this expression by taking out the common terms from the expression.
$ \Rightarrow 36{x^3} + 90{x^2} - 54x = 18x\left( {2{x^2} + 5x - 3} \right)$
Now, inside the bracket present in the RHS, we will do the middle term splitting. Therefore, we get
$ \Rightarrow 36{x^3} + 90{x^2} - 54x = 18x\left( {2{x^2} + 6x - x - 3} \right)$
$ \Rightarrow 36{x^3} + 90{x^2} - 54x = 18x\left[ {2x\left( {x + 3} \right) - 1\left( {x + 3} \right)} \right]$
Now, taking the brackets common, we get,
$ \Rightarrow 36{x^3} + 90{x^2} - 54x = 18x\left[ {\left( {2x - 1} \right)\left( {x + 3} \right)} \right]$
The number 18 in the RHS can be written as a product of its factor. So,
$ \Rightarrow 36{x^3} + 90{x^2} - 54x = \left( {2 \times 3 \times 3} \right)x\left[ {\left( {2x - 1} \right)\left( {x + 3} \right)} \right]$
Therefore, the factors of the second expression $36{x^3} + 90{x^2} - 54x$ are:
$\left( {2 \times 3 \times 3} \right),x,\left( {2x - 1} \right),\left( {x + 3} \right)$……………………………………$\left( 2 \right)$
Now, we are required to find the LCM of these two expressions.
In order to find the LCM, we take each factor present in each expression and if the factors are the same, then we take the largest possible power present in any one of the given two expressions and multiply it with the other factors to get the required LCM.
Hence, the LCM of these two expressions can be written as:
LCM$ = \left( {2 \times 2 \times 3 \times 3} \right) \times x \times \left( {x + 3} \right) \times \left( {{x^2} - 3x + 9} \right) \times \left( {2x - 1} \right)$
$ \Rightarrow $LCM$ = 36x\left( {2x - 1} \right)\left[ {\left( {x + 3} \right)\left( {{{\left( x \right)}^2} - \left( x \right)\left( 3 \right) + {{\left( 3 \right)}^2}} \right)} \right]$
Hence, again using the formula: $\left( {{a^3} + {b^3}} \right) = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)$, we get
$ \Rightarrow $LCM$ = 36x\left( {2x - 1} \right)\left( {{x^3} + {3^3}} \right) = 36x\left( {2x - 1} \right)\left( {{x^3} + 27} \right)$
Therefore, the required LCM of $12{x^4} + 324x$ , $36{x^3} + 90{x^2} - 54x$ is $36x\left( {2x - 1} \right)\left( {{x^3} + 27} \right)$.

Note: LCM is the least common multiple of given two integers such that, we take the smallest possible common factor of both the two integers and hence, we get the required LCM of the integers. But in the case of expressions, we find the product of the factors of the given expressions including the prime numbers if they are present. While multiplying the factors we must know that if a factor repeats itself in both the expressions then we take the one having the power higher than the other and multiply it with rest of the factors to get the required LCM of the given two expressions.