Question 1209731
<pre>

Couldn't find an online solver to help me much on this one.

{{{r+s+t = -matrix(1,3,coefficient,of,x^2)/matrix(1,3,coefficient,of,x^3) = -2}}}
{{{rs+rt+st = matrix(1,3,coefficient,of,x)/matrix(1,3,coefficient,of,x^3) = -5}}}
{{{rst = -matrix(1,2,constant,term)/matrix(1,3,coefficient,of,x^3) = -15}}}

{{{r+ s + t = -2}}}
{{{s+t=-2-r}}}
{{{r-2s-2t= r-2(s+t)=r-2(-2-r)=r+4+2r = 3r+4}}}

Write s for r, r for s, and t for t gives 

{{{s-2r-2t = 3t+4}}}

Write t for r, r for s, and s for t gives 

{{{t-2r-2s = 3s+4}}}

So we want:

{{{1/(3r+4)+1/(3s+4)+1/(3t+4)}}}

{{{((3s+4)(3t+4)+(3r+4)(3t+4)+(3r+4)(3s+4))/((3r+4)(3s+4)(3t+4))}}}

Multiply the numerator out:

{{{(3s+4)(3t+4)=9st+12s+12t+16=9st+12(s+t)+16}}}
Write r for s and t for t:
{{{(3r+4)(3t+4)=9rt+12r+12t+16=9rt+12(r+t)+16}}}
Write r for s and s for t
{{{(3r+4)(3s+4)=9rs+12r+12s+16=9rs+12(r+s)+16}}}

Add those three equations term by term:

{{{9(st+rt+rs)+12(2s+2t+2r)+48}}}
{{{9(st+rt+rs)+24(s+t+r)+48}}}
{{{9(rs+rt+st)+24(r+s+t)+48}}}
{{{9(-5)+24(-2)+48=-45-48+48 = -45}}}

That's the numerator.

Multiply the denominator out:

{{{(3r+4)(3s+4)(3t+4)=(3r+4)(9st+12s+12t+16)=27rst+36rs+36rt+36st+48r+48s+48t+64}}}

{{{27rst+36(rs+rt+st)+48(r+s+t)+64}}}

{{{27(-15)+36(-5)+48(-2)+64=-405-180-96+64=-617}}}

That's the denominator.

So the final answer is {{{(-45)/(-617)=45/617}}}

Edwin</pre>