Polynomial $a{x}^3+b{x}^2+cx+d=0$ has roots $x_1$, $x_2$ and $x_3$, then:

Attention to sign (−) alternating

a = (+), b = (−), c = (+), d = (−)

Example 01

Given an equation $x^3+5x^2+8x-4=0$ has roots $x_1$, $x_2$ and $x_3$.

Determine:

(A)   $x_1+x_2+x_3$

(B)   $x_1.x_2+x_1.x_3+x_2.x_3$

(C)   $x_1.x_2.x_3$

$x^3+5x^2+8x-4=0$

$a=1,b=5,c=8,d=-4$

Polynomial $a{x}^4+b{x}^3+c{x}^2+dx+e=0$ has roots $x_1$, $x_2$, $x_3$ and$x_4$, then:

Attention to sign (−) alternating

a = (+), b = (−), c = (+), d = (−), e = (+)

Example 02

Given an equation $x^4-2x^3-6x^2-7x-9=0$ has roots $x_1$, $x_2$, $x_3$ and $x_4$.

Determine:

(A)   $x_1+x_2+x_3+x_4$

(B)   $x_1.x_2+x_1.x_3+x_1.x_4+x_2.x_3+x_2.x_4+x_3.x_4$

(C)   $x_1.x_2.x_3+x_1.x_2.x_4+x_2.x_3.x_4$

(D)   $x_1.x_2.x_3.x_4$

$x^4-2x^3-6x^2-7x-9=0$

$a=1,b=-2,c=-6,d=-7,e=-9$