Question

Factors of $ 3{m^5} - 48m $ are:
A. $ 3m\left( {m - 1} \right)\left( {m - 3} \right) $
B. $ 3m\left( {m - 2} \right)\left( {m + 2} \right)\left( {{m^2} + 4} \right) $
C. $ 3m\left( {m - 1} \right)\left( {m - 2} \right)\left( {m + 1} \right) $
D. $ m\left( {m - 1} \right)\left( {m + 2} \right)\left( {{m^2} + 4} \right) $

Answer 1

Hint: In this type of problem, first take out the common value 3m from all the terms of the given expression then apply the identity of $ {a^2} - {b^2} $ .

Complete step-by-step answer:
The given expression for factorization is: $ 3{m^5} - 48m $
We have found that 3m is present in both terms of the given expression, so we can carry out 3m as common from both of them then we get,
 $ \Rightarrow 3m\left( {{m^4} - 16} \right) $
Here $ {m^4} $ is the square of $ {m^2} $ and 16 is the square of 4. So expressing the equation in form of $ {m^2} $ and 4 then we get,
 $ \Rightarrow \left( {{{\left( {{m^2}} \right)}^2} - {{\left( 4 \right)}^2}} \right) $
Now we found that the expression is similar to the identity $ {a^2} - {b^2} $ .
As we know that $ {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) $ .so, if we compare the expression with the identity then we get,
 $ a = {m^2},b = 4 $
So substituting the value of a and b in the given identity then we get,
 $
{\left( {{m^2}} \right)^2} - {\left( 4 \right)^2} = \left( {{m^2} + 4} \right)\left( {{m^2} - 4} \right)\\
 \Rightarrow {m^4} - 16 = \left( {{m^2} + 4} \right)\left( {{m^2} - 4} \right).........................(i)
 $
Here $ \left( {{m^2} - 4} \right) $ is also similar to the identity $ {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) $ so after comparing we get,
 $ a = m,b = 2 $
 $
{\left( m \right)^2} - {\left( 2 \right)^2} = \left( {m + 2} \right)\left( {m - 2} \right)\\
 \Rightarrow {m^2} - 4 = \left( {m + 2} \right)\left( {m - 2} \right).............................(ii)
 $
Hence substituting the value of $ {m^2} - 4 $ from the equation (ii) in equation (i) then we gets,
 $ \Rightarrow {m^4} - 16 = \left( {{m^2} + 4} \right)\left( {m + 2} \right)\left( {m - 2} \right) $
Substituting the value of $ {m^4} - 16 $ in the given expression then we get,
 $
 \Rightarrow 3m\left( {{m^4} - 16} \right) = 3m\left( {{m^2} + 4} \right)\left( {m + 2} \right)\left( {m - 2} \right)\\
 \Rightarrow 3{m^5} - 48m = 3m\left( {{m^2} + 4} \right)\left( {m + 2} \right)\left( {m - 2} \right)
 $
Hence the required factors of the given expression $ 3{m^5} - 48m $ is $ 3m\left( {{m^2} + 4} \right)\left( {m + 2} \right)\left( {m - 2} \right) $ .here, $ 3m,\left( {{m^2} + 4} \right),\left( {m + 2} \right) $ and $ \left( {m - 2} \right) $ are the factors of the given expression $ 3{m^5} - 48m $ .
So, the correct answer is “Option B”.

Note: This type of expression can also be factorise or the factor of the given expression can also be found by using the long division method. The identity of $ {a^2} - {b^2} $ is only applicable on the expression which has the terms as perfect square and also has subtraction operation between them.