Author Topic: At least one real zero?  (Read 6857 times)

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Offline Ender

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At least one real zero?
« on: April 10, 2008, 09:10:01 PM »
I like this problem 8)

Given that a + b/2 + c/3 + d/4 + e/5 = 0,

does the polynomial p(x) = a + bx + cx^2 + dx^3 + ex^4 have at least one real zero?

Feel free to post your progress / ask for hint.

Offline Nate

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Re: At least one real zero?
« Reply #1 on: April 10, 2008, 09:46:50 PM »
What are a,b,c,d,e?

Offline Joe

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Re: At least one real zero?
« Reply #2 on: April 10, 2008, 10:18:29 PM »
0 + 0/2 + 0/3 + 0/4 + 0/5 = 0

boom
I'd personally do as Joe suggests

You might be right about that, Joe.


Offline Sidoh

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Re: At least one real zero?
« Reply #3 on: April 10, 2008, 10:20:12 PM »
0 + 0/2 + 0/3 + 0/4 + 0/5 = 0

boom

He's asking for a general solution, not a solution of one in infinitely many cases. :P

Offline Ender

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Re: At least one real zero?
« Reply #4 on: April 10, 2008, 10:32:22 PM »
0 + 0/2 + 0/3 + 0/4 + 0/5 = 0

boom

The question is whether p(x) has at least one real zero.

You gave a solution to a + b/2 + c/3 + d/4 + e/5 = 0, i.e. (0, 0, 0, 0, 0), but this is entirely unrelated. You need to show that there is a solution to p(x) = 0.

And you also need not come up with an explicit solution. You can just as well show that there is one, without explicitly saying what this solution is.

Offline Nate

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Re: At least one real zero?
« Reply #5 on: April 10, 2008, 11:28:28 PM »
Im not going to bother but since the constraint seems to indicate P(1)=0, then I would say yea, there probably a zero of p(x)...too lazy to do the math to prove it though.

P(0)=0 as well but 0<x<1 for P(x) isn't zero...so its a parabola and the inflection point of that parabola is a zero of p(x) or something like that.
« Last Edit: April 10, 2008, 11:34:07 PM by Nate »

Offline Ender

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Re: At least one real zero?
« Reply #6 on: April 11, 2008, 12:00:35 AM »
Incorrect. p(1) is not necessarily 0, nor is p(0). In particular, p(1) = a + b + c + d + e, and p(0) = a. The constraint doesn't say that p(1) = 0.

Offline Nate

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Re: At least one real zero?
« Reply #7 on: April 11, 2008, 01:26:27 AM »
P(x) as in the anti derivative of p(x).  Your given constraint is P(1)=0.

Offline Ender

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Re: At least one real zero?
« Reply #8 on: April 11, 2008, 01:50:36 AM »
P(x) as in the anti derivative of p(x).  Your given constraint is P(1)=0.

Oh -- I didn't notice your notation.

You've made good progress, but it's not complete yet. P(0) = 0 and P(1) = 0. Now what?

Offline Nate

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Re: At least one real zero?
« Reply #9 on: April 11, 2008, 02:19:03 AM »
for 0 < x < 1, P(x) <> 0, therefore, p(x) needs to signs between 0 and 1 crossing the x-axis...I am sorry my math doesn't translate to words very well.

Offline Ender

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Re: At least one real zero?
« Reply #10 on: April 11, 2008, 02:38:05 AM »
Yep, that's basically it =) Though there's a particular calculus theorem that says what you just said succinctly.

Offline Ender

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Re: At least one real zero?
« Reply #11 on: April 11, 2008, 02:44:30 AM »
Forgot to say -- Good job, Nate!

Offline Ender

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Re: At least one real zero?
« Reply #12 on: April 11, 2008, 01:48:23 PM »
So yeah, now that Nate got it, I'll just say the calculus theorem. It's Rolle's Theorem.

Offline Nate

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Re: At least one real zero?
« Reply #13 on: April 11, 2008, 05:33:56 PM »
Calculus theorems are pointless, they can all be easily re derived on the spot if you don't see a solution intuitively.

Offline Newby

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Re: At least one real zero?
« Reply #14 on: April 11, 2008, 06:01:53 PM »
Calculus theorems are pointless, they can all be easily re derived on the spot if you don't see a solution intuitively.

Unless you're a math nerd like Ender or Rule, in which case they're amazing. :P
- Newby
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Quote
[17:32:45] * xar sets mode: -oooooooooo algorithm ban chris cipher newby stdio TehUser tnarongi|away vursed warz
[17:32:54] * xar sets mode: +o newby
[17:32:58] <xar> new rule
[17:33:02] <xar> me and newby rule all

I'd bet that you're currently bloated like a water ballon on a hot summer's day.

That analogy doesn't even make sense.  Why would a water balloon be especially bloated on a hot summer's day? For your sake, I hope there wasn't too much logic testing on your LSAT. 

Offline Ender

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Re: At least one real zero?
« Reply #15 on: April 11, 2008, 06:37:37 PM »
Calculus theorems are pointless, they can all be easily re derived on the spot if you don't see a solution intuitively.

An interesting thing about math is that an important theorem or principle may be obvious, but the way it changes your thinking can be profound. Take the pigeonhole principle, for example. It's ridiculously intuitive and obvious, but it can help you solve some really tough theorems or problems.
« Last Edit: April 11, 2008, 06:39:59 PM by Ender »

Offline rabbit

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Re: At least one real zero?
« Reply #16 on: April 11, 2008, 07:23:22 PM »
Calculus theorems are pointless, they can all be easily re derived on the spot if you don't see a solution intuitively.

An interesting thing about math is that an important theorem or principle may be obvious, but the way it changes your thinking can be profound. Take the pigeonhole principle, for example. It's ridiculously intuitive and obvious, but it can help you solve some really tough theorems or problems.

IIRC Sidoh disproved the pigeon hole principle.

Offline d&q

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Re: At least one real zero?
« Reply #17 on: April 11, 2008, 09:48:10 PM »
Some theorems are pretty cool and can be appreciated, but a theorem like Rolle's theorem is really intuitive and..bleh.
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Offline Ender

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Re: At least one real zero?
« Reply #18 on: April 12, 2008, 12:53:55 AM »
Some theorems are pretty cool and can be appreciated, but a theorem like Rolle's theorem is really intuitive and..bleh.

The whole point was that intuitive and obvious theorems and principles often deserve a lot of appreciation. MVT is very important :P