A polynomial $p(x)$ of degree n has n roots in the complex numbers. (see fundamental theorem of algebra).

If one of these roots is ${\alpha}$ , then $p({\alpha})=0$

Furthermore, if $p(x)=ax^n+bx^{n-1}+cx^{n-2}+...+px+q=0$ has roots ${\alpha},$ ${\beta},$ ${\gamma},$ ${\delta},$ ${\epsilon},....,$ ${\omega}$ then

• $\sum{\alpha}=-\frac{b}{a}$
• $\sum{\alpha}{\beta}=\frac{c}{a}$
• $\sum{\alpha}{\beta}{\gamma}=-\frac{d}{a}$
• $\sum{\alpha}{\beta}{\gamma}{\delta}=\frac{e}{a}$
• etc...
• ${\alpha}{\beta}{\gamma}{\delta}{\epsilon}...{\omega}=\pm\frac{q}{a}$
• If n is even then ${\alpha}{\beta}{\gamma}{\delta}{\epsilon}...{\omega}=\frac{q}{a}$
• If n is odd then ${\alpha}{\beta}{\gamma}{\delta}{\epsilon}...{\omega}=-\frac{q}{a}$

Moreover, if one of the roots of $p(x)=0$ is $z$ , where $z$ is complex (i.e. $z=x+iy$ and $y$ is non-zero) and all the coefficients $a,b,c...q$ are real numbers then the complex conjugate of $z$ is another root.
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