Question

Factorise the following $ 27{x^3} - 125{y^3} $

Answer 1

Hint: In the question, we need to factorise the given equation $ 27{x^3} - 125{y^3} $ .
Considering the fact that both the terms of the equation are perfect cubes, we will write them as mentioned below
 $
  27{x^3} = {(3x)^3} \\
  125{y^3} = {(5y)^3} \;
  $

Complete step-by-step answer:
Step 1 -To solve the equation, we will use the formula of difference of cubes identity: $ $
 $ {A^3} - {B^3} = (A - B)({A^2} + AB + {B^2}) $
Step 2- Let $ A = 3x,B = 5y $ $ $
Then,
$ 27{x^3} - 125{y^3} $
Step 3- Now, we will substitute the values of A and B I.e., 3x and 5y and then simplify the new equation as follows-
 $
   = {(3x)^3} - {(5y)^3} \\
   = (3x - 5y)({(3x)^2} + (3x)(5y) + {(5y)^2}) \\
   = (3x - 5y)(9{x^2} + 15xy + 25{y^2}) \;
  $
Hereby, we have factorised the equation given in the question by using the sum of cubes formula.
So, the correct answer is “ $ (3x - 5y)(9{x^2} + 15xy + 25{y^2}) $ ”.

Note: The equation $ (9{x^2} - 15xy + 25{y^2}) $ cannot be further factorised as it a multi-variable polynomial (is a finite sum of terms where each term looks like this (real#)(first variable)^(first whole#)(second variable)^(second whole)…(last variable)^(last whole#)).
Therefore in the above solution we have not solved it any further.