A community for students.
Here's the question you clicked on:
 0 viewing
zmudz
 one year ago
Given that \(x^n  (1/x^n)\) is expressible as a polynomial in \(x  (1/x)\) with real coefficients only if \(n\) is an odd positive integer, find \(P(z)\) so that \(P(x(1/x)) = x^5  (1/x)^5.\)
zmudz
 one year ago
Given that \(x^n  (1/x^n)\) is expressible as a polynomial in \(x  (1/x)\) with real coefficients only if \(n\) is an odd positive integer, find \(P(z)\) so that \(P(x(1/x)) = x^5  (1/x)^5.\)

This Question is Closed

myininaya
 one year ago
Best ResponseYou've already chosen the best response.0For some reason I first thought is to factor x^5(1/x)^5

myininaya
 one year ago
Best ResponseYou've already chosen the best response.0hmm I don't think I like that either I will continue to think

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.2If that helps: \[ x^5\left(x^5\frac{1}{x^5}\right)=x^{10}1 \]

myininaya
 one year ago
Best ResponseYou've already chosen the best response.0\[P(x\frac{1}{x})=x^5\frac{1}{x^5} \\ \text{ Let } z=x\frac{1}{x} \\ zx=x^21 \\ x^2zx1=0 \\ x=\frac{ z \pm \sqrt{z^2+4}}{2} \\ P(z)=(\frac{z+\sqrt{z^2+4}}{2})^5(\frac{2}{z+\sqrt{z^2+4}})^5\] I don't know which choice in x I should have made and P doesn't look like a polynomial but maybe it can be fix up I guess might be worth looking at what @thomas5267 said

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.2\[ P(x)=x^5+ax^4+bx^3+cx^2+dx+e\\ \left(x\frac{1}{x}\right)^5=x^55x^3+10x10x^{1}+5x^{3}x^{5}\\ \left(x\frac{1}{x}\right)^4=x^44x^2+64x^{2}+x^{4}\\ \left(x\frac{1}{x}\right)^3=x^33x+3x^{1}x^{3}\\ \left(x\frac{1}{x}\right)^2=x^22+x^{2}\\ x\frac{1}{x}\\ \left(x\frac{1}{x}\right)^5+a\left(x\frac{1}{x}\right)^4+b\left(x\frac{1}{x}\right)^3\\+c\left(x\frac{1}{x}\right)^2+d\left(x\frac{1}{x}\right)+e\\ =x^5+x^{5}\\ a=0\\ b=5\\ c=0\\ d=5\\ e=0 \] Inelegant, but it works.
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.