Atenção! Esta questão foi postada ha mais de 30 dias!
Exercício de Dispositivo de Briot-Ruffini e Teorema do Resto
Exercício:
P1(x) e P2(x) são polinômios do segundo grau que se anulam quando x=0.O resto da divisão de P1(x) por (x-1).(x+2) é 3x+1.O resto da divisão de P2(x) por (x+1).(x+2) é 2x-1.Então,o quociente da divisão de P1(x) por P2(x) é?
Pelo enunciado temos que, tanto P1(x) quanto P2(X), se anulam quando x=0, ou seja: P1(0) = P2(0) = 0
Sabemos que, quando dividimos P1(x) por (x-1)(x+2), temos como resto 3x+1.
Sabemos também que, em qualquer divisão:
Dividendo = Divisor x Quociente + Resto
Logo:
P1(X) = (x-1)(x+2) x Q1(X) + (3x+1)
A partir daí deduzimos que:
P1(-2) = -5
Do mesmo modo:
P2(x) = (x+1)(x+2) x Q2(x) + (2x-1)
Ou seja, P2(-2) = -5
Agora temos a seguinte divisão:
P1(x) = P2(x) x Q(x) + Resto
Pelo Teorema do Resto, e sabendo que P1(0) = P2(0) = 0, deduzimos que o resto da divisão P1(x) por P2(x) = 0
Outra informação dada pelo enunciado é que ambos os polinômios são do 2° grau, logo o quociente dessa divisão é um número real, e nao um polinômio.
Logo:
P1(x) = P2(x) x Q +0
Como P1(-2) = P2(-2) = -5:
P1(-2) = P2(-2) x Q
-5 = -5 x Q
Sendo assim:
Q = 1
4.2
Esta questão caiu nas Olimpiadas da Matemática 2009 segunda fase, tentei resolver pela igualdade de polinômios e não consegui.
(3x – m)2 + (4x – n)2 + (5x – 5 )2 = 2x, gostaria de saber do resultado da questão. Obrigado.
2-1=?
3
P1(X) = Q1(x-1)(x+2) + (3x+1), MAS P1(0)=0 DADO NA QUESTÃO, ENTÃO:
0=Q1(0-1)(0+2) + 3X0 +1=> Q1=1/2 ;
P2(X) = Q2(x+1)(x+2) + (2x+1), MAS P2(0)=0 DADO NA QUESTÃO, ENTÃO:
0=Q2(0+1)(0+2) + 2X0 +1=> Q2=1/2
ASSIM:
P1(X)=1/2(x-1)(x+2) + (3x+1) => P1(X)=X2/2 + 7X/2
e
P2(X)=1/2(x+1)(x+2) + (2x+1) => P2(X)=X2/2 + 7X/2
LOGO:
P1(X)/P2(X) =(X2/2 + 7X/2) / (X2/2 + 7X/2)
P1(X)/P2(X) = 1
vlw!!!
x2-7+12
I have to thank you for the efforts you’ve put in writing this blog.
I’m hoping to check out the same high-grade content by you later on as well.
In fact, your creative writing abilities has inspired me to get my own, personal site now
😉
I’ve been exploring for a bit for any high quality
articles or blog posts on this sort of area . Exploring in Yahoo I at last
stumbled upon this website. Studying this info So i’m glad to show that I’ve
a very just right uncanny feeling I came upon just what I needed.
I most indisputably will make certain to don?t omit this web site and provides it a glance on a relentless basis.
Keep on writing, great job!
This article is really a fastidious one it helps new
internet users, who are wishing in favor of blogging.
Thanks for sharing your thoughts. I truly appreciate your efforts and I am waiting for your further post thank you once
again.
We stumbled over here from a different web address and thought I may as well check things out.
I like what I see so i am just following you. Look forward
to looking at your web page yet again.
Hi! I know this is somewhat off topic but I was wondering if you knew where I could locate a captcha plugin for my comment form?
I’m using the same blog platform as yours and I’m
having trouble finding one? Thanks a lot!
Thanks for some other fantastic article. Where else may just anybody get that type of info in such a perfect way of writing?
I’ve a presentation next week, and I’m at the look for such info.
My programmer is trying to persuade me to move to .net from PHP.
I have always disliked the idea because of the costs.
But he’s tryiong none the less. I’ve been using Movable-type on numerous websites for about
a year and am worried about switching to another platform.
I have heard excellent things about blogengine.net.
Is there a way I can transfer all my wordpress posts into
it? Any kind of help would be really appreciated!
Fine way of describing, and good piece of writing to obtain data
regarding my presentation subject, which i am going to present in academy.
Fastidious response in return of this issue with solid arguments and describing everything about that.
I know this web site provides quality based articles or reviews and extra material,
is there any other website which provides these information in quality?
Hey, I think your blog might be having browser compatibility issues.
When I look at your website in Firefox, it looks fine
but when opening in Internet Explorer, it has some overlapping.
I just wanted to give you a quick heads up! Other then that, very good blog!