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If I have a polynomial $a_n x^n + a_{n-1}x^{n-1}+ \cdots + a_1 x + a_0$, and the roots of the polynomial is $r_1,r_2,\ldots,r_n$, then I can rewrite the polynomial as,

$a_n x^n + a_{n-1}x^{n-1} \cdots + a_1 x + a_0 = (x-r_1)\cdots(x-r_n)$. Now if I wanted to express each coefficient of the polynomial by its roots, how would I go about that?

For example, is $a_0 = (-1)^n(r_1\cdots r_n)$? What about $a_1,a_2,\ldots,a_n$?

Would $a_1 = (-1)^{n-1}(r_1 r_2\cdots r_{n-1} + r_1 r_2\cdots r_{n-2} r_n + \cdots)$?

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  • $\begingroup$ ...you can write the polynomial as $$a_nx^n+\ldots+a_1x+a_0=\color{red}{a_n}(x-r_1)\cdot\ldots\cdot(x-r_n)$$ $\endgroup$ – Timbuc Jan 28 '15 at 18:32
  • $\begingroup$ The following link might be useful: math.stackexchange.com/questions/69544/… $\endgroup$ – Krish Jan 28 '15 at 18:32
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The answer to both questions is yes, provided that $a_n=1$.

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