Can anybody help me with a problem regarding partial fractions?

Question

I have been asked to express 10+5x-8x^2 over (2-3x^2)(4+x) in terms of its partial fractions, but I have been coming up with ridiculus answers and have no idea what I am doing wrong.

Answer 1

Check out that website. The very last problem on the page is similar to the one you are working. Remember with partial fractions if your denominator is a quadratic, then your numerator is going to have to be linear (Ax + B) not just a coefficient (A). Also remember that you can use the method of comparing coefficients to find your partial fractions. Work the problem again and see if you match my answer.

I got:

(7X -23 / 2-3x^2) + ( 5 / 4+x )

Answer 2

(10+5x-8x^2)/(2-3x^2)(4+x) = A/(sqrt(2/3)+x) + B/(sqrt(2/3)-x) + C(4+x)

A(sqrt(2/3)-x)(4+x) + B(sqrt(2/3)+x)(4+x) + C(sqrt(2/3)+x)(sqrt(2/3)-x) = (10+5x-8x^2)

The rest is incredibly ugly but here's the gist: when you completely expand the left side of the equation, you'll end up with terms that have x^2, x, and no x. Make seperate equations with like terms so that you end up with three equations and three unknows. And now shove them in a calculator to solve for the system of equations.

Good luck

Answer 3

I don't think you are doing anything wrong. Perhaps the convention of writing the known first and unknown later is a bit awkward (at least to me).

I have included a few references just in case you may want to look at them.

According to ref #2 you should get:
-(x+1)/(3x^2 -2) + 3/(x+4)

Answer 4

10+5x-8x^2/(2-3x^2) (4+x) = A/ (2-3x^2) + B/ (4+x)

10+5x-8x^2 = A(4+x) + B(2-3x^2)

10+5x-8x^2 = A * 4 + A * x + 2B - B *3*x^2

10+5x-8x^2 = A 4 + A x + 2B - 3Bx^2

I think you have to factor 10+5x-8x^2

[You have to do this step]

Try to equate co-efficiants for 'x'

?10+5x-8x^2? = 4A + [A x] + 2B - [3Bx^2]

?10+5x-8x^2? = A + (-3B)

Then, co-efficiant for constants:

?10+5x-8x^2? = 4A + 2B

now the simultanious equation...

?10+5x-8x^2? = A + (-3B) ---- (1)
?10+5x-8x^2? = 4A + 2B ---- (2)

Furthermore, you can be ending up with a:

-(x+1)/(3x^2 -2) + 3/(x+4)