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:
Here, we need to factorise the given expression. We will complete the square in the given expression, and apply the algebraic identity for the square of the sum of two numbers. Then, we will use the algebraic identity for the product of the sum and difference of two numbers to factorise the given expression and find the correct option.

Formula Used:
We will use the following formulas:
1) The square of the sum of two numbers is given by the algebraic identity \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\].
2) The product of the sum and difference of two numbers is given by the algebraic identity \[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\].

Complete step by step solution:
We have to factorise the expression \[{x^4} + {x^2}{y^2} + {y^4}\].
We will complete the square in the expression to rewrite \[{x^4} + {x^2}{y^2} + {y^4}\].
To complete the square, we need to add and subtract the term \[{x^2}{y^2}\] in the expression \[{x^4} + {x^2}{y^2} + {y^4}\].
Adding and subtracting \[{x^2}{y^2}\] in the expression \[{x^4} + {x^2}{y^2} + {y^4}\], we get
\[ \Rightarrow {x^4} + {x^2}{y^2} + {y^4} = {x^4} + {x^2}{y^2} + {y^4} + {x^2}{y^2} - {x^2}{y^2}\]
Simplifying the expression, we get
\[ \Rightarrow {x^4} + {x^2}{y^2} + {y^4} = {x^4} + 2{x^2}{y^2} + {y^4} - {x^2}{y^2}\]
Rewriting the terms as squares of some number, we get
\[ \Rightarrow {x^4} + {x^2}{y^2} + {y^4} = {\left( {{x^2}} \right)^2} + 2\left( {{x^2}} \right)\left( {{y^2}} \right) + {\left( {{y^2}} \right)^2} - {\left( {xy} \right)^2}\]
The square of the sum of two numbers is given by the algebraic identity \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\].
Substituting \[a = {x^2}\] and \[b = {y^2}\] in the algebraic identity \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\], we get
\[ \Rightarrow {\left( {{x^2} + {y^2}} \right)^2} = {\left( {{x^2}} \right)^2} + 2\left( {{x^2}} \right)\left( {{y^2}} \right) + {\left( {{y^2}} \right)^2}\]
Substituting \[{\left( {{x^2}} \right)^2} + 2\left( {{x^2}} \right)\left( {{y^2}} \right) + {\left( {{y^2}} \right)^2} = {\left( {{x^2} + {y^2}} \right)^2}\] in the right hand side of the equation \[{x^4} + {x^2}{y^2} + {y^4} = {\left( {{x^2}} \right)^2} + 2\left( {{x^2}} \right)\left( {{y^2}} \right) + {\left( {{y^2}} \right)^2} - {\left( {xy} \right)^2}\], we get
\[ \Rightarrow {x^4} + {x^2}{y^2} + {y^4} = {\left( {{x^2} + {y^2}} \right)^2} - {\left( {xy} \right)^2}\]
The product of the sum and difference of two numbers is given by the algebraic identity \[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\].
Using the algebraic identity \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\] in the right hand side of the equation \[{x^4} + {x^2}{y^2} + {y^4} = {\left( {{x^2} + {y^2}} \right)^2} - {\left( {xy} \right)^2}\], we get
\[ \Rightarrow {x^4} + {x^2}{y^2} + {y^4} = \left( {{x^2} + {y^2} + xy} \right)\left( {{x^2} + {y^2} - xy} \right)\]
Therefore, we have factored the algebraic expression \[{x^4} + {x^2}{y^2} + {y^4}\] as \[\left( {{x^2} + {y^2} + xy} \right)\left( {{x^2} + {y^2} - xy} \right)\].

Thus, the correct option is option (d).

Note:
We factorised the given algebraic expression in the solution using algebraic identities. Factorisation is the process of writing an equation as a product of its factors. We factored the algebraic expression \[{x^4} + {x^2}{y^2} + {y^4}\] as \[\left( {{x^2} + {y^2} + xy} \right)\left( {{x^2} + {y^2} - xy} \right)\]. This means that \[\left( {{x^2} + {y^2} + xy} \right)\] and \[\left( {{x^2} + {y^2} - xy} \right)\] are the factors of the algebraic expression \[{x^4} + {x^2}{y^2} + {y^4}\].