Question

Factorise: $ x - 8x{y^3} $
A. $ x\left( {1 - 2y} \right)\left( {1 + 2y + 4{y^2}} \right) $
B. $ x\left( {1 + 2y} \right)\left( {1 + 2y + 4{y^2}} \right) $
C. $ x\left( {1 - 2y} \right)\left( {1 - 2y + 4{y^2}} \right) $
D. $ x\left( {1 + 2y} \right)\left( {1 - 2y + 4{y^2}} \right) $

Answer 1

Hint: Check the common between both terms of a given expression, then take out this common term and apply the identity of $ {a^3} - {b^3} $ on the remaining expression to get the factored form of the given expression.

Complete step-by-step answer:
The given expression for factorisation is $ x - 8x{y^3} $
We have found that x is present in both terms of the expression hence we take out x common from the whole expression then we get,
 $ \Rightarrow x\left( {1 - 8{y^3}} \right) $
Now, we will see the remaining expression as $ \left( {1 - 8{y^3}} \right) $ .
We know that the cube of 1 is 1 and the cube of 2 is 8.
This expression can also be written as: $ \left( {{{\left( 1 \right)}^3} - {{\left( {2y} \right)}^3}} \right) $ , which is similar to the identity of $ {a^3} - {b^3} $ and we know that its expanded form is $ {a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right) $
If we compare the right-hand side of the identity with $ \left( {1 - 8{y^3}} \right) $ then we will get,
 $ a = 1,b = 2y $
So we will substitute the value of a and b in the left hand side of the identity too then we get,

 $ {\left( 1 \right)^3} - {\left( {2y} \right)^3} = \left( {1 - 2y} \right)\left( {{{\left( 1 \right)}^2} + 1 \times 2y + {{\left( {2y} \right)}^2}} \right) $
After simplifying the equation we get,
 $ 1 - 8{y^3} = \left( {1 - 2y} \right)\left( {1 + 2y + 4{y^2}} \right) $
Now we will substitute the simplified form of $ 1 - 8{y^3} $ in the given expression then we get,
 $
 \Rightarrow x\left( {1 - 8{y^3}} \right)\\
 \Rightarrow x\left( {1 - 2y} \right)\left( {1 + 2y + 4{y^2}} \right)
 $
This is the required factorise form of the given expression.
Hence, $ x\left( {1 - 2y} \right)\left( {1 + 2y + 4{y^2}} \right) $ are the factors of the given expression $ x - 8x{y^3} $
So, the correct answer is “Option B”.

Note: In case if you don’t know the identity of $ {a^3} - {b^3} $ then you may also use the identity of $ {\left( {a - b} \right)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3} $ then take the values of $ {a^3} - {b^3} $ on one side of equation and remaining equation on another side of the equation. This expression can also be factorised by using the long division method.