f derivative divides f, when?

ricsi046 said:
can you give me any hint regarding this question?
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Um... You'd kinda have to post a question first. Only the above text is displaying within your post.

Or are you referring to your subject line? If so, what is the full and exact text of the exercise? What are the instructions? What have you tried? Where are you stuck?

Please be complete. Thank you! ;)
 
What do you mean by "divides"? That is usually used in the sense of "divides evenly" but that only makes sense for "discrete" things like integers or polynomials.
 
This is the full question
When can a polynomial be divided with it's derivative?
We got this question on a classical algebra course,we have been learning about polynomials in the past weeks.So i think f is some polynomial from C[x]
As for what i tried,well i think that f derivative's roots must be the roots of f too,with the same or higher multiplicity.But i don't know what could i do with that fact.

EDIT:By divide i mean that for example 4 divides 12( 4|12),i thought that was a correct word,sorry if not
 
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So you mean "when can a polynomial be evenly divided by its derivative?" Think about writing a general polynomial as a product of linear terms: \(\displaystyle p(x)= (x- a)^n(x- b)^m\cdot\cdot\cdot(x- c)^p\) with as many terms as you have distinct zeros. By the product rule, \(\displaystyle p'= n(x- a)^{n-1}(x- b)^m\cdot\cdot\cdot(x- c)^p+ m(x- a)^n(x- b)^{m-1}\cdot\cdot\cdot (x- c)^p+ \cdot\cdot\cdot +p(x- a)^n(x- b)^m\cdot\cdot\cdot (x- c)^{p-1}\).
 
ricsi046 said:
When can a polynomial be divided with [its] derivative?

got this question [in] a classical algebra course

By divide i mean that for example 4 divides 12 (4|12)

thought that was a correct word,sorry if not
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Hi ricsi:

The highlighted phrases above (red and blue) do not have the same meaning; it's a matter of syntax. :)

I just noticed your subject line; that informal wording is better than the red version above. (I had skipped your OP the first time because you didn't say anything about function f.)

We could also post
Under what conditions is f'(x)|f(x) true, when f(x) is a polynomial?

When is a polynomial function f(x) divisible by f'(x)?

​

For me, the answer to "when can a polynomial be divided by its derivative" is "whenever the polynomial is not a constant". (That is, there's nothing to stop us from dividing a polynomial by its derivative, unless the derivative is zero.)

Anyway, you've already reasoned out one particular type of polynomial which satisfies f'(x)|f(x), yes?

If not, you should play around with some very basic polynomials. The particular polynomial-type will be obvious, once you realize it.

After that, ponder other types of polynomials -- along the lines of what hallsofIvy posted.

Ciao :cool:

PS: I've never seen a classical algebra course get into derivatives; good for you. (I haven't even seen many precalculus courses that get to derivatives.)
 
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