How do you factor \[1-256{{y}^{8}}\]?

Hint: The algebraic expression of the form \[{{a}^{2}}-{{b}^{2}}\] is called difference of squares. The factored form of these types of expression is \[\left( a+b \right)\left( a-b \right)\]. The factored form of a polynomial expression is used to find the roots of the polynomial. We should know that there is no factorization for expressions that are of the form of addition of squares.

Complete step by step answer:
The given expression is \[1-256{{y}^{8}}\]. It has two terms; the first term is 1 and the second term is \[256{{y}^{8}}\].
As, we know that 1 is square of itself, the second term can also be written as \[{{\left( 16{{y}^{4}} \right)}^{2}}\]. Using this simplification in the given expression, it can be written as \[1-256{{y}^{8}}={{1}^{2}}-{{\left( 16{{y}^{4}} \right)}^{2}}\].
As we can see that this expression is evaluating the difference between two square terms, it is the difference of square form. We know that the difference of square expression \[{{a}^{2}}-{{b}^{2}}\] is factorized as \[\left( a+b \right)\left( a-b \right)\]. Here, we have a, and b are 1 and \[16{{y}^{4}}\] respectively. Substituting the values in the expansion, we get
\[1-256{{y}^{8}}=\left( 1+16{{y}^{4}} \right)\left( 1-16{{y}^{4}} \right)\]
Here the first-factor term is \[1+16{{y}^{4}}\]. As, we know that it can not be further factorized, as it has no real roots. The second-factor term is \[1-16{{y}^{4}}\]. We know that 16 is the square of 4. Hence, we can express it as \[{{1}^{2}}-{{\left( 4{{y}^{2}} \right)}^{2}}\]. This is also of the form of difference of squares. It’s expansion will be \[\left( 1+4{{y}^{2}} \right)\left( 1-4{{y}^{2}} \right)\]. Thus, the expansion of the given expression can be written as \[1-256{{y}^{8}}=\left( 1+16{{y}^{4}} \right)\left( 1+4{{y}^{2}} \right)\left( 1-4{{y}^{2}} \right)\].
The last factor can be factorized as \[\left( 1+2y \right)\left( 1-2y \right)\].

Thus, the factorization of given expression is \[1-256{{y}^{8}}=\left( 1+16{{y}^{4}} \right)\left( 1+4{{y}^{2}} \right)\left( 1+2y \right)\left( 1-2y \right)\]

Note: The maximum number of factors a polynomial can have equals to its degree. This will happen only if all the roots of the polynomial are real. In this example, we get three factors but the degree of this expression is four.