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title: Solving polynomials formally

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# Solving polynomials formally

Formally solving a polynomial $P(X)=0$ means to find a number $x$ such that $P(x)=0$ but for any lower degree polynomial $Q$ we have $Q(x)\neq 0$.

For example, let us formaly solve the polynomial $X^2-1=0$. It seems $X=1$ or $X=-1$. But **none of them** solve the polynomial formally. Because although $(-1)^2-1=0$, but $-1$ also satiesfies $-1+1=0$. This means $-1$ is both a root for $X^2-1$ and for $X+1$. By solving the polynomial formally we do not wish the root comes from lower-degree polynomials.

In other words, **it is impossible to formally solve a polynomial of degree 2 or larger by old numbers**. We must introduce new numbers and solve it.

> **Ex**: Formally solve the polynomial equation $X^2-1=0$ 
> We introduce new number $\mu$ generated by $X^2-1$. Clear $\mu$ is a root of this polynomial. We also discover $-\mu$ also satisfies 
> $$(-\mu)^2-1=\mu^2-1=0$$. So $\mu$ and $-\mu$ formally solve the polynomial equation $X^2-1=0$ because $a\mu+b\neq0$ for all real numbers $a,b$.