Question

Factorise: $ {a^4} - {b^4} $

Answer 1

Hint: In this type of question, factorise the expression by making terms of the expression perfect square and then by using the identity of $ {x^2} - {y^2} $ . This identity is applicable for the equation having subtraction between the values which are perfect squares.

Complete step-by-step answer:
The given expression for factorisation is $ {a^4} - {b^4} $ .
Here we see that $ {a^4} $ and $ {b^4} $ have the exponent of 4. To make the expression as quadratic as well make the terms perfect square we can write the given expression in this way:
 $ {\left( {{a^2}} \right)^2} - {\left( {{b^2}} \right)^2} $
The given expression has the degree of 2, as the highest exponent in the expression is 2. Now we check the end terms of the expression which are $ {\left( {{a^2}} \right)^2} $ and $ {\left( {{b^2}} \right)^2} $ .
Here, we found that $ {\left( {{a^2}} \right)^2} $ is the perfect square of \[{a^2}\] but $ {\left( {{b^2}} \right)^2} $ is a perfect square of $ {b^2} $ , hence by using the identity of $ {x^2} - {y^2} $ to find the factor of the given expression.
So, as we know that,
 $ {x^2} - {y^2} = \left( {x + y} \right)\left( {x - y} \right) $
Comparing the given expression with the identity then we get,
 $ x = {a^2},y = {b^2} $
Substituting the values of x and y in the identity then we get,
 $
\Rightarrow {\left( {{a^2}} \right)^2} - {\left( {{b^2}} \right)^2} = \left( {{a^2} + {b^2}} \right)\left( {{a^2} - {b^2}} \right)\\
\Rightarrow {a^4} - {b^4} = \left( {{a^2} + {b^2}} \right)\left( {{a^2} - {b^2}} \right)
 $
Now, again apply the identity $ {x^2} - {y^2} $ for $ \left( {{a^2} - {b^2}} \right) $ . On comparing the expression with identity then we get,
 $ x = a,y = b $
After substituting the values we get,
 $ {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) $
Substituting the value of $ \left( {{a^2} - {b^2}} \right) $ in the above expression then we get,
 $ \Rightarrow {a^4} - {b^4} = \left( {{a^2} + {b^2}} \right)\left( {a + b} \right)\left( {a - b} \right) $
Therefore, $ \left( {{a^2} + {b^2}} \right)\left( {a + b} \right)\left( {a - b} \right) $ is the required factor of the given expression $ {a^4} - {b^4} $ and $ {a^4} - {b^4} $ completely divisible by these factors.

Note: This expression can also be factorised by using the middle term splitting method and then carry out common from the given expression. Here, $ {x^2} - {y^2} $ identity is applicable on the equation having only perfect squares.