<s>[INST] Polynomials - The sum of two roots If the sum of two roots of
$$x^4  + 2x^3 - 8x^2 - 18x - 9 = 0$$ is
$0$, find
the roots of the equation
 [/INST] Let those two roots be $a,-a$ and the other two roots are $b,c$
$$\text{Now, }(x-a)(x+a)(x-b)(x-c)=(x^2-a^2)\{x^2-(b+c)x+bc\}$$
$$=x^4-x^3(b+c)+(bc-a^2)x^2+a^2(b+c)x-a^2bc$$
Comparing the coefficients of $x^3$ of this with that of $x^4  + 2x^3 - 8x^2 - 18x - 9 = 0,$
$b+c=-2$
Comparing the coefficients of $x,a^2(b+c)=-18\implies 2a^2=18\implies a=\pm3$
Comparing the  constants, $a^2bc=9\implies bc=1$
So, $b,c$ are the roots of the equation $t^2-(-2)t+(1)=0\implies t=-1,-1$
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