Question

Factors of $ {a^4} - 2{a^2}{b^2} + {b^4} $ are:
A. $ {\left( {a + b} \right)^2} $
B. $ \left( {{a^2} + {b^2} + 2ab} \right) $
C. $ \left( {{a^2} + {b^2} - 2ab} \right) $
D. $ \left( {a - b} \right) $

Answer 1

Hint: To find the factor of the given expression $ {a^4} - 2{a^2}{b^2} + {b^4} $ make the whole expression in form of identity $ {\left( {x - y} \right)^2} = \left( {{x^2} - 2xy + {y^2}} \right) $ and then simplified form will be the factor of the given expression.

Complete step-by-step answer:
Here, the given expression for factorization is: $ {a^4} - 2{a^2}{b^2} + {b^4} $
We found that $ {a^4} $ is the perfect square of $ {a^2} $ and the term $ {b^4} $ is also the perfect square of $ {b^2} $ . So we should express the given expression in form of $ {a^2} $ and $ {b^2} $ then we get,
 $ \Rightarrow {\left( {{a^2}} \right)^2} - 2{a^2}{b^2} + {\left( {{b^2}} \right)^2} $
Hence the expression is similar with the identity $ {\left( {x - y} \right)^2} = \left( {{x^2} - 2xy + {y^2}} \right) $ . We compare the expression with the identity then we get,
 $ a = {a^2},b = {b^2} $
So substituting the value of a and b in the given identity, the result of which is as shown,
 $
\left( {{{\left( {{a^2}} \right)}^2} - 2{a^2}{b^2} + {{\left( {{b^2}} \right)}^2}} \right) = {\left( {{a^2} - {b^2}} \right)^2}\\
 \Rightarrow \left( {{a^4} - 2{a^2}{b^2} + {b^4}} \right) = \left( {{a^2} - {b^2}} \right)\left( {{a^2} - {b^2}} \right).........................(i)
 $
Here $ \left( {{a^2} - {b^2}} \right) $ is also similar to the identity $ {x^2} - {y^2} = \left( {x + y} \right)\left( {x - y} \right) $ so after comparing we get,
 $ x = a,y = b $
 $
{\left( a \right)^2} - {\left( b \right)^2} = \left( {a + b} \right)\left( {a - b} \right)\\
 \Rightarrow {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right).............................(ii)
 $
Hence substituting the value of $ {a^2} - {b^2} $ from the equation (ii) in equation (i) then we gets,
 $ \Rightarrow \left( {{a^4} - 2{a^2}{b^2} + {b^4}} \right) = \left( {a - b} \right)\left( {a + b} \right)\left( {a - b} \right)\left( {a + b} \right) $
This expression can also be written as:
 $
\Rightarrow \left( {{a^4} - 2{a^2}{b^2} + {b^4}} \right) = {\left( {a - b} \right)^2}{\left( {a + b} \right)^2}\\
\Rightarrow \left( {{a^4} - 2{a^2}{b^2} + {b^4}} \right) = \left( {{a^2} + 2ab + {b^2}} \right)\left( {{a^2} - 2ab + {b^2}} \right)
 $
Hence the required factors of the given expression $ \left( {{a^4} - 2{a^2}{b^2} + {b^4}} \right) $ is $ \left( {{a^2} + 2ab + {b^2}} \right)\left( {{a^2} - 2ab + {b^2}} \right) $ .

Here, option A is correct as $ {\left( {a + b} \right)^2} $ is the factor of given expression.
Option B is correct as $ \left( {{a^2} + 2ab + {b^2}} \right) $ is the factor of given expression.
Option C is correct as $ \left( {{a^2} - 2ab + {b^2}} \right) $ is the factor of given expression.
Option D is correct as $ \left( {a - b} \right) $ is the factor of given expression.

Note: This type of expression can also be factorised or the factor of the given expression can also be found by using the long division method. The identity of ${a^2} - {b^2}$is only applicable on the expression which has the terms as perfect square and also have subtraction operation between them.