Question

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

Answer 1

Hint: In this type of question make the expression quadratic by adding and subtracting the $ {x^2}{y^2} $ in the given expression then use identity $ {\left( {a + b} \right)^2} $ and $ {a^2} - {b^2} $ to get the factorise form of the given expression.

Complete step-by-step answer:
The given expression is $ {x^4} + {x^2}{y^2} + {y^4} $ .
In the given expression adding and subtracting $ {x^2}{y^2} $ then we get,
 $
{x^4} + {x^2}{y^2} + {y^4}\\
 \Rightarrow {x^4} + {x^2}{y^2} + {y^4} - {x^2}{y^2} + {x^2}{y^2}\\
 \Rightarrow {x^4} + 2{x^2}{y^2} + {y^4} - {x^2}{y^2}\\
 \Rightarrow \left( {{x^4} + 2{x^2}{y^2} + {y^4}} \right) - {x^2}{y^2}\\
 \Rightarrow \left( {{{\left( {{x^2}} \right)}^2} + 2{x^2}{y^2} + {{\left( {{y^2}} \right)}^2}} \right) - {x^2}{y^2}
 $
The given expression has the degree of 4, as the highest exponent in the expression is 4. Now if we look at the expression within the bracket then we find that the expression is of quadratic form.
Now, checking the end terms of the quadratic expression then, we found that $ {x^4} $ is the perfect square of {x^2} but $ {y^4} $ is a perfect square $ {y^2} $ so; we will use identity to solve the expression.
As we know that: $ {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2} $
Comparing the Left-hand part from the quadratic part of the given expression then we get,
 $ a = {x^2},b = {y^2} $
So substituting the identity in the given expression then we get,
 $
 \Rightarrow {\left( {{x^2} + {y^2}} \right)^2} - {x^2}{y^2}\\
 \Rightarrow {\left( {{x^2} + {y^2}} \right)^2} - {\left( {xy} \right)^2}
 $
Same exponent over the value which is in multiple then this exponent should be taken as common over the product of both values.
As we know that: $ {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) $
If we compare the expression with this identity we get,
 $ a = {x^2} + {y^2},b = xy $
So substituting the values in the identity then we get,
 $ \Rightarrow \left( {{x^2} + {y^2} - xy} \right)\left( {{x^2} + {y^2} + xy} \right) $
This is the required factorise form of the given expression.

So, the correct answer is “Option D”.

Note: You can also do it by using the splitting method or taking common values. Using identity increases the accuracy of the solution. Use the identity after comparing them with the given expression.