Factorization of a polynomial

Question

I need to find the roots of this polynomial $$2x^2-x^4-x=0.$$ I know that it is necessary the factorization to obtain $$-x(x-1)(x^2+x-1)=0.$$ I asked to factorize my polynomial to Mathematica. The last step to find the roots is to split the polynomial in three equations.

What kind of method of factorization could I use to do things like this? I also know that there are a lot of methods of factorization; could you suggest me the most useful, simplest to start with?

Thank you.

Answer 1

You have already solved most of this by yourself. If you have a high-degree polynomial (say $\deg P > 2$), try to find some roots by "guessing", for example $x=0$ and $x=1$ in your polynomial. Finding a root, to get another root, you do polynomial (long) division: $$p(x) = 2x^2 - x^4 - x \stackrel!= 0$$ $x=0$ was found, so get $$p_1(x) := \frac{p(x)}{x} = 2x - x^3-1\stackrel!=0$$ Guessing another root, $x=1$, you get $$p_2(x) := \frac{p_1(x)}{x-1} = -x^2 -x +1 \stackrel!=0$$ From there ($\deg p_2 \le 2$) we can continue by completing the square: $$-(x^2 + x + \frac14 - \frac54) \stackrel!=0$$ Thus $$(x+\frac12)^2 \stackrel!= \frac54$$ $$\Rightarrow x \stackrel!= -\frac12 \pm \frac{\sqrt5}2$$ So your roots are: $$\{x | p(x) = 0\} = \{0\} \cup \{x|p_1(x) = 0\} = \{0,1\} \cup \{x|p_2(x) = 0\} = \{0,1,-\frac12+\frac{\sqrt5}2, -\frac12 -\frac{\sqrt5}2\}$$

Answer 2

After factoring out $x$ you obtain a cubic whose highest and lowest coefficients are $\color{#c00}{\pm1}.\,$ If it is reducible then it has a rational root, so by the Rational Root Test, that root can only be $\color{#c00}{\pm1}$. Thus for polynomials of this form you need only evaluate it a two points to complete the factorization.

Generally one can use the Rational Root Test to reduce the factorization of cubics to the testing of a finite number of rational roots, but there will be more than $2$ possible roots to check when the polynomial's high and low coeff's have many factors.

More generally, one can reduce polynomial factorization to the integer factorization of a few values taken by the polynomial, using an old algorithm due to Bernoulli, Schubert, Kronecker, Hausman. However, this is not efficient compared to modern algorithmic methods.