simplify: 5/x^3 - 5/(x + h)^3/h. I will attempt to write it using equation in the next post. the h is supposed to be under the whole of the rest of the equation.

\[5divx ^{3} - 5\div(x + h)^{3}divh\] I'm not sure how to do it as an equation.

\[(5/x^3 - 5/(x+h)^3)/h\] ?

Yes

ok so the h is under everything

or is the h only under the last one

under the whole thing. Like with the brackets, that is how I should have done it.

ok

So keep the h out and do it last, and simplify the 2 other terms first by finding the least common denominator and then multiplying everything out and then simplifying again

Step 1: Predend the h isn't there and do it last. Step 2. Find the least common denominator for the two terms. Step 3: Combine terms. Step 4: Multiply out the numerator and re-factor as much as you can.

The least common denominator? 5/x^3 - 5/(x^3 + 2x^2h + 3xh^2 + h^3) h^3?

or x^3 because there is one on both sides?

the least common denominator is (x^3)(x+h)^3. I'd keep it like that for now so it doesn't get too messy too soon.

lol this is already messy to me! okay... 5(x + h)^3/(x^3)(x + h)^3 - 5(x)^3/(x + h)^3(x)^3 ? so then I can combine 5(x + h)^3 - 5x^3/x^3(x + h)^3 ?

(5(x + h)^3 - 5x^3)/x^3(x + h)^3

Yes that is correct.

then I get, after cancelling 5x^3, (15x^2h + 15xh^2 + 5h^3)/(x^3 + 3x^2h + 3xh^2 + h^3)(x^3) and now I'm stuck again.

is x^3 * x^3 = x^6?

oh hang on, would it have been better to leave it intact?

no perhaps because then I would have to divide 5x^3 by the same.

Yeah I don't think you will need to expand the denominator; the only reason you would ever expand the denominator is if you thought you could expand and then factor the denominator differently such that you factor out a term from the numerator that could cancel with a term in the denominator.

So I'd leave it in tact for now

So, 5 - 5x^3/((x + h)^3)(x^3) ?

right now I've got the same as you've got: [15hx^2 + 15h^2x + 5h^3]/[(x^3)(x+h)^3] and that is after factoring and canceling that h from the beginning

I'm trying to see if that can be simplified further.

Woops that was before factoring out that h

So then it becomes: (15x^2 + 15xh + 5h^2) / [(x^3)(x+h)^3]

good :) I was wondering how you got that answer after the h factor so you can remove a h from each part, both top and bottom?

oh, no, just the numerator at the very top?

Yeah only the numerator

Wow thanks so much for taking the time to help! This is so complicated, really appreciate it!

No problem. I think this is as simplified as it can get, as the roots of the numerator appear to be imaginary numbers.

Imaginary? lol okay :)

Oh I just read this: For polynomials in more than one variable the notion of root does not exist. So I guess it's as simplified as it can get.

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