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What is the LCM of $\left( {6{x^3} + 60{x^2} + 150x} \right)$ and $\left( {3{x^4} + 12{x^3} - 15{x^2}} \right)$?
A.$6{x^2}{(x + 5)^2}(x - 1)$
B.$3{x^2}{(x + 5)^2}(x - 1)$
C.$6{x^2}{(x + 5)^2}{(x - 1)^2}$
D.$3{x^2}{(x + 5)^2}{(x - 1)^2}$

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Last updated date: 20th Jun 2024
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Answer
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Hint: At first we need to factorize the given polynomials into their simpler forms, and from that we can write the lcm . To find the LCM , at first we need to write all the elements but in the case of common terms we consider the term with the highest degree

Complete step-by-step answer:
Step-1
The first polynomial given is $\left( {6{x^3} + 60{x^2} + 150x} \right)$.
Now we need to factorize this polynomial into product of linear polynomials to find the LCM
Let's first start with the numerical value 6, taking 6 as a common element we get,
$ \Rightarrow 6\left( {{x^3} + 10{x^2} + 25x} \right)$
The next common element is x .So taking x common we get,
$ \Rightarrow 6x\left( {{x^2} + 10{x^{}} + 25} \right)$
Now we need to factorize the quadratic polynomial $\left( {{x^2} + 10{x^{}} + 25} \right)$
Now by splitting the middle term,we get
$ \Rightarrow 6x\left( {{x^2} + 5x + 5x + 25} \right)$
Now taking x common from the first two terms and 5 common in the next two terms we get,
$ \Rightarrow 6x\left( {x(x + 5) + 5(x + 5)} \right) \\
   \Rightarrow 6x(x + 5)(x + 5) \\
   \Rightarrow 6x{(x + 5)^2} \\
$
Step 2 :
Now the second polynomial is $\left( {3{x^4} + 12{x^3} - 15{x^2}} \right)$
Now we need to factorize this polynomial into product of linear polynomials to find the LCM
Let's first start with the numerical value 3, taking 3 as a common element we get,
$ \Rightarrow 3\left( {{x^4} + 4{x^3} - 5{x^2}} \right)$
The next common element is ${x^2}$ .So taking ${x^2}$common we get,
$ \Rightarrow 3{x^2}\left( {{x^2} + 4x - 5} \right)$
Now we need to factorize the quadratic polynomial $\left( {{x^2} + 4x - 5} \right)$
Now by splitting the middle term,we get
$ \Rightarrow 3{x^2}\left( {{x^2} + 5x - x - 5} \right)$
Now taking x common from the first two terms and - 1 common in the next two terms we get, $3{x^2}(x + 5)(x - 1)$


Step 3 :
Now to find the LCM , at first we need to write all the elements but in the case of common terms we consider the term with the highest degree.
The given polynomials are $6x{(x + 5)^2}$ and $3{x^2}(x + 5)(x - 1)$
At first we need to find the LCM of 3 and 6
The LCM of 3 and 6 is 6
And now we have $x$ and .${x^2}$. So the lcm is the one with the highest degree , (i . e ) ${x^2}$
Then we have $x + 5$ and ${(x + 5)^2}$ and consider the one with the highest degree we select .${(x + 5)^2}$.
And including $x - 1$
The LCM is $6{x^2}{(x + 5)^2}(x - 1)$

The correct option is A.

Note: Many students forget to find the lcm of the numerical values and write only the algebraic terms which is wrong
Additional information
For any two polynomials p(x) and q(x)
LCM * GCD = p(x) * q(x)