I have a very little background on mathematics. I have a very basic question about solving polynomial equations.
if we have $$P_n(x) = 0$$ where $P_n(x)$ is a polynomial of degree $n$.For example: $$x^5-3x^3+x^2-7x+3 = 0$$
Why we cannot just decompose the polynomial equation to a system of equations solving them separately and the solutions to the original polynomial will be the intersection of the roots of each equation:
$$\begin{cases}
x^5-3x^3 = 0 \\ \text{and} \\x^2-7x+3=0
\end{cases}$$
There is a lot of possibilities (infinity?) to choose this sub-equations.
I know this seems very stupid but could you tell me which rule of logic I broke.
As far as I know, if $a=0$ and $b=0$ then we can add them to form $a+b=0$ we cannot go the other way, that is $a+b=0\implies a=0 \quad\text{and}\, b=0$ why?
Best Answer
We have that: $$a +b =0 \to a=-b$$ Which means it holds for $a=b=0$, but also that you can't go the other way, as you said.
Your system of equations should be: $$\begin{cases} x^5-3x^3 = k \\ \text{and} \\x^2-7x+3=-k \end{cases}$$
We can solve $x^2-7x+(3+k)=0$ for $x=\frac{7\pm\sqrt{37-4k}}{2}$, and then insert than into the quintic for solutions, but this results in:
$$\frac{(7\pm\sqrt{37-4k})^5}{32}-\frac{3(7\pm\sqrt{37-4k})^3}{8}-k=0$$
One question springs to mind, is this really any simpler? To my mind, not at all.
We can substitute $t=7+\sqrt{37-4k}$, but this requires the solution of:
$$t^5-12t^3+8t^2-112t+96=0$$ which is more complicated than the polynomial we are trying to solve.
All in all: is this a valid method? Absolutely. Is it practical? Absolutely not in my opinion.