Resolve the factor ${x^4} + 16{x^2} + 256$

Hint: We need to use the algebraic identity ${a^4} + {a^2}{b^2} + {b^4}$ to factorize the given expression. Here the degree of all the terms in the given expression is four. We cannot find any term whose power is less than four. On factoring the expression ${a^4} + {a^2}{b^2} + {b^4}$ we get the factors ${a^2} + ab + {b^2}$ and ${a^2} - ab + {b^2}$ . We have to use this identity to factorize the given expression by substituting the values respectively.

Complete answer:
We have an expression ${x^4} + 16{x^2} + 256$.
When we multiply back the factors of an algebraic expression we get back the expression itself. From the basic definition of factors, we know that to find the factors of an algebraic expression we have to express the expression as a product of two or more expressions. We are given an expression with all the terms having a power of four. So, the degree of the expression is four.
We compare the given equation with the identity ${a^4} + {a^2}{b^2} + {b^4}$ and get:
$a = x,b = 4$
So we substitute the terms in the given identity.
So, on factorizing the given expression we get:
${x^4} + 16{x^2} + 256$
Writing them in powers of four.
$ = {(x)^4} + 16{x^2} + {(4)^4}$
Factorizing the expression by using the identity we get:
$ = ({x^2} + {4^2} + 4x)({x^2} + {4^2} - 4x)$
Simplifying the expressions we get:
$ = ({x^2} + 16 + 4x)({x^2} + 16 - 4x)$
Rearranging the terms we get:
$ = ({x^2} + 4x + 16)({x^2} - 4x + 16)$
Hence, the required factorization of ${x^4} + 16{x^2} + 256$ is $({x^2} + 4x + 16)({x^2} - 4x + 16)$ .

Note:
These expressions are not factorizable further. If the expressions are to be factorizable we need the middle terms to be multiplied by $2$ further. Only then we can use the identities ${(a + b)^2} = {a^2} + 2ab + {b^2}$ and ${(a - b)^2} = {a^2} - 2ab + {b^2}$ to factorize the expressions further. This identity is only used for solving expressions where all the terms have even powers. This identity cannot be applied to expressions having odd powers.