Question

Write the partial fraction decomposition of the rational expression. Check your result algebraically.

(x^2+6x+3)/(x^3+x)

Answer 1

Let's break up your expression into partial fractions. Noting that the denominator factors into \[x(x^2+1)\]

Answer 2

explain please? i have exam tomorrow

Answer 3

yes i got that part

Answer 4

We get

\[\frac{ x^2+6x+3 }{ x^3+x }=\frac{ A }{ x }+\frac{ Bx+C }{ x^2+1 }\]

Answer 5

yes!

Answer 6

Before we move on, please ask any questions that you may have.

Answer 7

i understand so far! thank you for helping

Answer 8

Good! Now we'd like to eliminate the fractions.

Look at the "A" term. Missing from its denominator is (x^2+1), RIGHT?

so, multiply both A and x by (x^2+1). OK?

Answer 9

ummmmmmmmmmmmm

Answer 10

oh right

Answer 11

i get it, multiply A and x by (x^2+1) and x with the other term

Answer 12

sorry if i am wrong

Answer 13

Just for verification, we need to obtain the following:\[\frac{ A(x^2+1) }{ x(x^2+1) }\]

Answer 14

Can you now do the B term yourself? Mult. num. and denom. each by x.

Answer 15

yeah

Answer 16

wouldn't be x^3 +x then for denominator?

Answer 17

after multiplying by x

Answer 18

Yes, but there's no point in doing this multiplication. We're trying to get rid of the denom.

Answer 19

oh so it should be (Bx+C)(x)

Answer 20

As a matter of fact, we should now have \[x^2+6x+3=A(x^2+1)+Bx\]

Answer 21

with no denominators. Can you believe that? Yes, sorry, x(Bx +C).

Answer 22

yes!

Answer 23

i see where i went wrong but i am on track

Answer 24

Good. Finish this on your own and let me know if and when you have further questions. Glad to see you again on OpenStudy.

Answer 25

hey thanks

Answer 26

My pleasure!