<s>[INST] How to factorize polynomials to the 5th degree? I have the polynomial:
$$2x^5-x^4+10x^3-5x^2+8x-4$$
and I know that the final result is:
$$(2x-1)(x^4+5x^2+4) = (2x-1)(x^2+1)(x^2+4)$$
But how would you do it step by step? I've seen a couple of videos and blogs about it, but they mostly use examples, where their is a common factor between the expressions but in this case their are none.
 [/INST] The question slightly wrong in my opinion. Please try multiplying the factors or the eqn. and lets see if you will get back the original question.
The correct equation is $P(x): 2x^5-x^4-10x^3+5x^2+8x-4$
Group and Factorize: $(2x^5-x^4) (-10x^3+5x^2) (8x-4) =0$
                 $x^4(2x-1)   -5x^2(2x-1)  4(2x-1)$

        Factor out $(2x-1)$ then we have $x^4-5x^2+4$

Our first factor of the equation is $(2x-1)$
This looks like a quadratic form but our first coefficient has 4th degree
look let's expand $x^4-5x^2+4$ and we have $(x^2-4)$ and $(x^2-1)$. $-4$ and $-1$ when added will be $-4-1 = -5$ and when multiply $-4x-1 = 4. x^2 x x^2 = x^4$
so its just splitting them and distributing them on $-4$ and $-1$ which are the product of the constant and sum of the middle term.
Now with $(x^2-4)$ and $(x^2-1)$ we can have $(x-2) (x+2)$ and $(x-1) (x+1)$.
so our final factors of the polynomial are $(2x-1) (x-2) (x+2) (x-1) (x+1)$
Therefore; the zeros of the polynomial are $1/2, 2, -2, 1, -1$.
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