Question

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

Answer 1

Hint: Given an expression. Rewrite the equation in the form of the difference of the square of two terms. Then, apply the formula of the difference of squares of two terms. Then, again rewrite the equation. Again, apply the formula of the difference of squares of two terms. Repeat the factorization process, till the expression is simplified.

Formula used:
The formula for the difference of squares of two terms is given by:
${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$

Complete step-by-step answer:
We are given the expression ${x^8} - 256$. First, we will rewrite the expression as a difference of squares of two terms.
$ \Rightarrow {\left( {{x^4}} \right)^2} - {\left( {16} \right)^2}$
Now, we will apply the formula for the difference of squares of two terms to the expression.
$ \Rightarrow \left( {{x^4} - 16} \right)\left( {{x^4} + 16} \right)$
Now, the expression $\left( {{x^4} + 16} \right)$ cannot be simplified further. Then, we will factorize the term $\left( {{x^4} - 16} \right)$ by first rewriting the expression as a difference of squares of two terms.
$ \Rightarrow \left( {{x^4} - 16} \right) = {\left( {{x^2}} \right)^2} - {\left( 4 \right)^2}$
Now, we will apply the formula for the difference of squares of two terms to the expression.
$\left( {{x^2} - 4} \right)\left( {{x^2} + 4} \right)$
Now, the expression $\left( {{x^2} + 4} \right)$ cannot be simplified further. Then, we will factorize the term $\left( {{x^2} - 4} \right)$ by first rewriting the expression as a difference of squares of two terms.
$ \Rightarrow \left( {{x^2} - 4} \right) = {\left( x \right)^2} - {\left( 2 \right)^2}$
Now, we will apply the formula for the difference of squares of two terms to the expression.
$\left( {x - 2} \right)\left( {x + 2} \right)$
Now, write the factors of the expression by combining all factors together.
$\left( {x - 2} \right)\left( {x + 2} \right)\left( {{x^2} + 4} \right)\left( {{x^4} + 16} \right)$

Thus, the factors of the expression are $\left( {x - 2} \right)\left( {x + 2} \right)\left( {{x^2} + 4} \right)\left( {{x^4} + 16} \right)$

Note:
In such types of questions students mainly don't get an approach on how to solve it. In such types of questions, students are mainly confused about which algebraic identity must be applied. Students mainly forget to convert the expression as a difference of the square of two numbers or terms.