TITLE: Factorization of polynomials with degree higher than 2
QUESTION [5 upvotes]: I need help to factorize $x^4-x^2+16$. I have tried to take $x^4$ as $(x^2)^2$ and factorize it in the typical way of factorizing a quadratic expression but that did not help. Can someone help me to factor this and also introduce me to the procedure that i need to follow to factorize expressions with degree higher than two?

REPLY [2 votes]: Can be done by a simple completion of squares. The presence of $x^4$ and 16 hints towards a possibility of a $(x^2+4)^2$ so just calculate that and compare it with your question function. $(x^2+4)^2 = x^4+8x^2+16$ which leaves us with a difference only in the $x^2$ term. This difference turns out to be $9x^2$. So our function reduces to $$(x^2+4)^2-(3x)^2$$
It's easy to see after this, a simple use of $a^2-b^2 = (a+b)(a-b)$ gives the final answer to be $$(x^2-3x+4)(x^2+3x+4)$$