Question

Factorize: \[{x^4} - 625\].

Answer 1

Hint: Use the algebraic identity  \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\] to find the factors.

The given algebraic expression in the question is in the degree of 4 so by using the algebraic expression we will reduce the given expression to the degree of 2 and then again by using the algebraic identity we will find the factors of the given expression in the question.

Complete step-by-step answer:

Given the expression whose factor is to be found is \[{x^4} - 625\].

In the given expression we can say the degree of the expression is 4 and the expression contains two terms with the operator ‘-‘ between them.

Now we can also write the given expression as  \[{\left( {{x^2}} \right)^2} - {\left( {{5^2}} \right)^2}\] 

We know the algebraic identity \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\] , so by using this identity we can write the given expression as 

 \[{\left( {{x^2}} \right)^2} - {\left( {{5^2}} \right)^2} = \left( {{x^2} - {5^2}} \right)\left( {{x^2} + {5^2}} \right)\] 

Now again we will use the algebraic identity to expand \[\left( {{x^2} - {5^2}} \right)\] , hence we can further write the obtained expression as

 \[{\left( {{x^2}} \right)^2} - {\left( {{5^2}} \right)^2} = \left( {{x^2} - {5^2}} \right)\left( {{x^2} + {5^2}} \right) = \left( {x - 5} \right)\left( {x + 5} \right)\left( {{x^2} + {5^2}} \right)\] 

Hence we can say the factors of the given algebraic expression  \[{x^4} - 625\] are \[\left( {x - 5} \right)\] , \[\left( {x + 5} \right)\] and  \[\left( {{x^2} + {5^2}} \right)\] 

So, the correct answer is “\[\left( {x - 5} \right)\] , \[\left( {x + 5} \right)\] and  \[\left( {{x^2} + {5^2}} \right)\] ”.

Note: It is interesting to note that whenever an expression is split into their factors, if we multiply those factors together we will get the expression whose factors were found, this method is used to verify whether the factor for the expression is correct or not.

To verify this, if we multiply the obtained multiple together we get \[\left( {x - 5} \right)\left( {x + 5} \right)\left( {{x^2} + {5^2}} \right) = \left( {{x^2} - {5^2}} \right) \left( {{x^2} + {5^2}} \right) = {\left( {{x^2}} \right)^2} - {\left( {{5^2}} \right)^2}\] , hence we can say the obtained answer was correct.