Question

What are the factors of $ 54{a^3} - 250{b^3} $ ? ( Contains multiple correct options )
A) $ 2 $
B) $ 3a - 5b $
C) $ 5a - 3b $
D) $ 9{a^2} + 15ab + 25{b^2} $

Answer 1

Hint: For the given expression, we can use the formula of $ {\left( {x - y} \right)^3} $ . This formula can be written as $ {\left( {x - y} \right)^3} = {x^3} - 3{x^2}y + 3x{y^2} - {y^3} $ . From this we can write the formula for $ {x^3} - {y^3} $ as $ {x^3} - {y^3} = \left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right) $

Complete step-by-step answer:
Given to us an expression $ 54{a^3} - 250{b^3} $
Firstly we can take the number $ 2 $ common in this expression so this expression can now be written as $ 2\left( {27{a^3} - 125{b^3}} \right) $
Since the given expression is a multiple of two, the number $ 2 $ is a factor of this expression.
Now, in this expression we know that $ 27 $ is a cube of three and $ 125 $ is the cube value of five. So now this expression can be written as follows.
$\Rightarrow 2\left( {{3^3}{a^3} - {5^3}{b^3}} \right) = 2\left( {{{\left( {3a} \right)}^3} - {{\left( {5b} \right)}^3}} \right) $
Now, this is in the form of $ {x^3} - {y^3} $
The formula for this can be written as
$\Rightarrow {x^3} - {y^3} = \left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right) $
The values of x and y from the given expression are $ x = 3a $ and $ y = 5b $
By substituting these values in the formula, we get
 $ 2\left( {{{\left( {3a} \right)}^3} - {{\left( {5b} \right)}^3}} \right) = 2\left[ {\left( {3a - 5b} \right)\left( {{{\left( {3a} \right)}^2} + 3a \times 5b + {{\left( {5b} \right)}^2}} \right)} \right] $
On solving, we get
$\Rightarrow 2\left( {3a - 5b} \right)\left( {9{a^2} + 15ab + 25{b^2}} \right) $
By multiplying the three terms $ 2,3a - 5b $ and $ 9{a^2} + 15ab + 25{b^2} $ in this expression the resultant would be our given expression which is $ 54{a^3} - 250{b^3} $
Therefore the factors of the given expression are $ 2,3a - 5b $ and $ 9{a^2} + 15ab + 25{b^2} $ i.e. Options A, B and D.
So, the correct answer is “Option A,B and D”.

Note: It is to be noted that the given expression $ 54{a^3} - 250{b^3} $ is a multiple of all the three of its factors calculated i.e. $ 2,3a - 5b $ and $ 9{a^2} + 15ab + 25{b^2} $ . Not that we can also modify the formula of $ {\left( {x - y} \right)^3} $ to find the value of the given expression which is in the form of $ {x^3} - {y^3} $ . We can write this modification as follows.
From $ {\left( {x - y} \right)^3} = {x^3} - 3{x^2}y + 3x{y^2} - {y^3} $ we get $ {x^3} - {y^3} = {\left( {x - y} \right)^3} + 3{x^2}y - 3x{y^2} $ .