Question

How do you factorize${x^8} - 256$?

Answer 1

Hint: In this question, we have been asked to factorize the given expression. First, simplify the second term of the question such that it can be written as a power of$8$. Then, use the identity of difference of squares to simplify the given expression. You will have to apply this identity twice or thrice. Apply until there is no difference of squares left. Then, arrange the terms in ascending order of their powers. You will get your answer.

Formula used:
${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$

Complete step by step answer:
We have been given an expression and we have to write its factors.
$ \Rightarrow {x^8} - 256$ …. (Given)
Now, we know that $256 = {16^2}$ .
Substituting this in the given expression –
$ \Rightarrow {\left( {{x^4}} \right)^2} - {\left( {16} \right)^2}$
Now, we will use the identity of${a^2} - {b^2}$, where we will assume $a = {x^4}$ and $b = 16$ .
Using the identity${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ ,
Therefore, ${\left( {{x^4}} \right)^2} - {\left( {16} \right)^2} = \left( {{x^4} + 16} \right)\left( {{x^4} - 16} \right)$ .
Hence, we have got $\left( {{x^4} + 16} \right)\left( {{x^4} - 16} \right)$ ,
$ \Rightarrow \left( {{x^4} + 16} \right)\left( {{x^4} - 16} \right)$
Now, we will again expand the second term. Before that, we knew $16 = {4^2}$. Putting this in the equation,
$ \Rightarrow \left( {{x^4} + 16} \right)\left( {{{\left( {{x^2}} \right)}^2} - {4^2}} \right)$
Again, using the identity of ${a^2} - {b^2}$, where we will assume $a = {x^2}$ and $b = 4$ .
Using the identity ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ ,
$ \Rightarrow \left( {{x^4} + 16} \right)\left( {{x^2} + 4} \right)\left( {{x^2} - 4} \right)$
There is still some scope of expanding the above equation. We know that $4 = {2^2}$. Putting in the above equation,
$ \Rightarrow \left( {{x^4} + 16} \right)\left( {{x^2} + 4} \right)\left( {{x^2} - {2^2}} \right)$
Again, using the identity of${a^2} - {b^2}$, where we will assume $a = x$ and $b = 2$ .
Using the identity${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ ,
$ \Rightarrow \left( {{x^4} + 16} \right)\left( {{x^2} + 4} \right)\left( {x + 2} \right)\left( {x - 2} \right)$
Rearranging the terms on the basis of power,
\[ \Rightarrow \left( {x - 2} \right)\left( {x + 2} \right)\left( {{x^2} + 4} \right)\left( {{x^4} + 16} \right)\]

Hence, ${x^8} - 256 = \left( {x - 2} \right)\left( {x + 2} \right)\left( {{x^2} + 4} \right)\left( {{x^4} + 16} \right)$

Note: While solving the above question, students always forget the two main things. They are as follows:
We have to find the factors and not the value of the variable. If you solve using a quadratic formula, you have to find the factors because the formula gives us the values. If you are using the middle term method, you will get the factors directly.
Manu students end their question once they have found the values of x or any other variable that you assumed. The question asked the value of y and not the value of the assumed variable. So, in the end, remember to find the values of a given variable.