<s>[INST] How are the sum and product of root formulas derived? For a polynomial of degree n,say 
$$F(x)=a_{0}+a_{1}x+a_{2}x^2+a_{3}x^3+....+a_{n}x^n$$
The sum of roots is $-\frac{a_{n-1}}{a_n}$ and the product of roots is $(-1)^n \frac{a_{0}}{a_n}$ where the $a_i%$ are the coefficients of the polynomial. I've been using these formulas for some problems, but I don't get why they are true. How are they derived? Thanks.
 [/INST] $P(x)=a_nx^n+a_{n-1}x^{n-1}+a_1x+a_0=a_n(x-x_1)(x-x_2)\cdot \cdot (x-x_n)$ 
To show the product of roots:
Need to calculate $P(0)$
$P(0)=a_n0^n+a_{n-1}0^{n-1}+a_10+a_0=a_0$
$P(0)=a_n(0-x_1)(0-x_2)\cdot \cdot (0-x_n)=a_n(-x_1)(-x_2)\cdot \cdot (-x_n)=a_n(-1)x_1(-1)x_2\cdot \cdot (-1)x_n=a_n(-1)^nx_1x_2\cdot \cdot x_n$ 
$P(0)=a_n(-1)^nx_1x_2\cdot \cdot x_n=a_0$
$x_1x_2\cdot \cdot x_n=\frac{a_0(-1)^n}{a_n}$
To show Sum of roots:
Need to focus the coefficient of $x^{n-1}$
$(x-x_1)(x-x_2)=x^2-(x_1+x_2)x+x_1x_2$
$(x-x_1)(x-x_2)(x-x_3)=(x^2-(x_1+x_2)x+x_1x_2)(x-x_3)=x^3-(x_1+x_2)x^2-x_3x^2 \cdot \cdot=x^3-(x_1+x_2+x_3)x^2 \cdot \cdot$
$(x-x_1)(x-x_2)(x-x_3)(x-x_4)=(x^3-(x_1+x_2+x_3)x^2 \cdot \cdot)(x-x_4)=x^4-(x_1+x_2+x_3)x^3-x_4x^3 \cdot \cdot=x^4-(x_1+x_2+x_3+x_4)x^3 \cdot \cdot\cdot \cdot$
$a_n(x-x_1)(x-x_2)\cdot \cdot (x-x_n)=a_n(x^n-(x_1+x_2+x_3+x_4+\cdot \cdot+x_n)x^{n-1} \cdot \cdot \cdot \cdot )=a_nx^n-a_n(x_1+x_2+x_3+x_4+\cdot \cdot+x_n)x^{n-1} \cdot \cdot \cdot \cdot=a_nx^n+a_{n-1}x^{n-1}+a_1x+a_0$ 
If we equal the coefficient of $x^{n-1}$  the result
 is: $-a_n(x_1+x_2+x_3+x_4+\cdot \cdot+x_n)=a_{n-1}$
$x_1+x_2+x_3+x_4+\cdot \cdot+x_n=\frac{-a_{n-1}}{a_n}$
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