Question

Write the following expression as the sum of three partial fractions:

Answer 1

\[E = \frac{ 2x - 5x^2 }{ (x+1)^3 }\]

Answer 2

Hmm try
\[\frac{A}{x+1}+\frac{B}{(x+1)^2}+\frac{C}{(x+1)^3}\]

Answer 3

Ok lemme try that and get back to you.

Answer 4

Alright

Answer 5

but won't I have to take x as -1 all the time?

Answer 6

But then A and B and C will always come as 0. Can I take any value for x and then solve them simultaneously??

Answer 7

You multiply the whole thing by \((x+1)^3\), you get
\[2x-5x^2=A(x+1)^2+B(x+1)+C\]

Answer 8

\[ \frac{ 2x - 5x^2 }{ (x+1)^3 }= \frac{A}{x+1}+\frac{B}{(x+1)^2}+\frac{C}{(x+1)^3} \\ \\ 2x-5x^2=A(x+1)^2+B(x+1)+C\]

Answer 9

Wait wait hold on. Does the equation become

A(x+1)^2 + B(x+!) + C = 2x - 5x^2

Answer 10

Yes correct

Answer 11

Then maybe we'll comapare coefficients for this,
\[x^2:~~~ -5=A \\ \\ x:~~~2=2A+B \\ \\ Constant:~~~0=A+B+C\]

Answer 12

Wait, we can do that?? I thought we need to put in values for x and then solve.

Answer 13

You can compare them straight away

Answer 14

So far i've got C = -1 and A+B = 7

Answer 15

But you can't compare them, on the left it's (x+1)^2 and on the right it's just x^2. That won't make sense, would it.

Answer 16

Oh but wait, you accounted for that. Sorry :P

Answer 17

How dow you get that?
\[-5x^2=Ax^2 \\ \\ 2x=2Ax+Bx \\ \\ 0=A+B+C\]

Answer 18

Yeah

Answer 19

A will be -5
Then you can find B and C already

Answer 20

Yup, i've got 12 and -7.

Answer 21

correct

Answer 22

just a note :
you can also *mix* the 2 methods,
if you put x=-1, you directly get C
then go for comparison :)

Answer 23

Ok, and the next pert of the question says check that you've got it right. How do I do that??

Answer 24

Aw

Answer 25

Aw?? :P

Answer 26

*part.

Answer 27

I'll make common denominators for that :P, but I'm sure there's another method, just can't think of it right now

Answer 28

Meh. @hartnn , got anything?

Answer 29

common denominator is the best(and maybe the only) way

Answer 30

So i make the partial fractions into one fraction again??

Answer 31

Yes

Answer 32

Dear lord. Why does my math teacher do this to me, WHYYY

Answer 33

like, he makes me expand in into three, and then says make it one again
ughh

Answer 34

That's how you check your work to make sure you expanded it correctly :P

It is tedious, though.

Answer 35

Ikr??