OpenStudy (tylerd):

partial fraction decomposition

3 years ago
OpenStudy (tylerd):

\[\frac{ -9x^2-28x+12 }{ (x+4)(x^2+3x+6) }=\frac{ f(x) }{ x+4 }+\frac{ g(x) }{ x^2+3x+6 }\]

3 years ago
OpenStudy (anonymous):

You can use a constant function for \(f\) and a linear one for \(g\).

3 years ago
OpenStudy (tylerd):

ya

3 years ago
OpenStudy (tylerd):

and that is because something to do with the discriminant correct?

3 years ago
OpenStudy (tylerd):

brb 1-2 mins

3 years ago
OpenStudy (anonymous):

Well you can use the discriminant of the quadratic factor to see if it's possible to reduce it further, but that's not the case here since the discriminant is negative.

3 years ago
OpenStudy (tylerd):

hmm........

3 years ago
OpenStudy (tylerd):

ya.......

3 years ago
OpenStudy (anonymous):

So do you know how to proceed from here if you let \(f(x)=A\) and \(g(x)=Bx+C\)?

3 years ago
OpenStudy (tylerd):

ya thats where i started, i ended up with a bunch of messy algebra

3 years ago
OpenStudy (anonymous):

If you have access to the program, Mathematica, the function "Apart" saves a lot of work and time to deal with a problem like this one.\[\text{Apart}\left[\frac{-9 x^2-28 x+12}{(x+4) \left(x^2+3 x+6\right)}\right] \]\[\left\{0.000539,\frac{6-7 x}{x^2+3 x+6}-\frac{2}{x+4}\right\} \] 539 millions of a second on a mid 2010 IMac.

3 years ago
OpenStudy (tylerd):

well i suppose having the answer would be nice, but, i got an exam on this tomorrow, and no comps/cellphones allowed...

3 years ago
OpenStudy (anonymous):

\[\begin{align*} \frac{ -9x^2-28x+12 }{ (x+4)(x^2+3x+6) }&=\frac{A}{ x+4 }+\frac{Bx+C}{ x^2+3x+6 }\\\\ -9x^2-28x+12 &=A(x^2+3x+6)+(Bx+C)(x+4)\\\\ &=(A+B)x^2+(3A+4B+C)x+6A+4C \end{align*}\] Matching up coefficients gives \[\begin{cases} A+B=-9\\ 3A+4B+C=-28\\ 6A+4C=12 \end{cases}\]

3 years ago
OpenStudy (tylerd):

ah i see i factored it differently on the 3rd line there.

3 years ago
OpenStudy (tylerd):

and ended up going in circles with the algebra

3 years ago
OpenStudy (tylerd):

@SithsAndGiggles how did you get A+B=-9 and 3A+4B+C=-28

3 years ago
OpenStudy (tylerd):

think i figured it out nvm..

3 years ago