Math guys, how did this question yield this answer?

Math Experts need your help wrote:

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

I've been trying to work out how the above problem yields this: (x^3+3x+2)/(x^3)(x+1)^2.

It doesn't. Plug in x = 1 for example.

Your solution should be negative.

You are doing subtraction, so figure out the lowest common denominator. Luckily, in this problem, you only have to convert the first fraction. The second one is good to go. Then subtract, following the rules of subtracting fractions. Factor the numerator. Expand the binomial in the denominator, factor out the GCF, and then simplify. It works out the way you typed it except that your answer should be negative, like I said above.

Just followed this and I figured it out, damn it. Thank you Jamin, and esp. Ozzie for the extra advice!

Math Experts need your help wrote:

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

I've been trying to work out how the above problem yields this: (x^3+3x+2)/(x^3)(x+1)^2. Would appreciate any help!

It doesn't, the sign is wrong.

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

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

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

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

etc etc

I just noticed that too, mighty appreciated Wigins. Thanks again everyone!

Ozzie wrote:

Your solution should be negative.

You are doing subtraction, so figure out the lowest common denominator. Luckily, in this problem, you only have to convert the first fraction. The second one is good to go. Then subtract, following the rules of subtracting fractions. Factor the numerator. Expand the binomial in the denominator, factor out the GCF, and then simplify. It works out the way you typed it except that your answer should be negative, like I said above.

Big mistake I made was not expanding the denominator, really appreciate the tip on that man.

You got this from taking the derivative of f(x) = (x^2 + 1) / (x^3 + x^2).

The point of the problem was probably to solve using partial fractions.

f(x) = A/(x+1) + B/x + C/x^2.

Make a common denominator and get

f(x) = ( A x^2 + B x (x+1) + C (x+1) ) / (x^3 + x^2)

Solve ( A x^2 + B x (x+1) + C (x+1) ) = x^2 + 1 for A, B, C.

It's easy: C = 1, because that's the only constant term.

B = -1, because that's the only linear term left.

A = 2.

So f(x) = 2 / (x+1) - 1/x + 1/x^2.

Derivative of this last form is more straight forward than the full quotient rule, less multiplying out random polynomials and hoping not to make a mistake..

Also, it is obvious from the partial fractions method that the denominator of the derivative should be (a divisor of) x^3 (x+1)^2, because that is the LCM of the denominators of the derivatives of the terms.