Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

Callisto

  • 2 years ago

Formulas and Identities #1 Question 31 Consider the identity \(x^{99}-1 ≡ (x^2-1)P(x) + Cx+D\), where P(x) is a polynomial in x. Find the values of C and D

  • This Question is Closed
  1. dpaInc
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    C=1, D=-1 ???

  2. Callisto
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Bingo ... How did you get it?!?

  3. dpaInc
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    silly me... i wrote the proof of it on the margin of my notebook and now i can't find it....:(

  4. dpaInc
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    substitute x=1 , and you'll get the equation: 0 = C + D substitute x=-1, and you'll get the equation: -2 = -C + D now solve the system...

  5. Callisto
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    ..... Silly me....

  6. yakeyglee
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Well, I don't really see an easy way to do this, so I guess I'll attempt it by "brute force". Let us generically define \(P(x)\) as follows. I think it's pretty clear that it will be of order 97\[P(x) = a_0 + a_1 x + a_2 x^2 + \cdots a_{97} x^{97}\]Multiplying \(x^2-1\) gives us the following.\[(x^2-1)P(x) = x^2P(x)-P(x) =\]\[= -a_0 - a_1 x + (a_0 - a_2)x^2 + \cdots (a_{95}-a_{97})x^{97}-a_{96}x^{98}-a_{97}x^{99} \]Now, let's add \(Cx+D\).\[x^{99}-1=(C-a_0) + (D-a_1) x + (a_0 - a_2)x^2 + \cdots (a_{95}-a_{97})x^{97}-a_{96}x^{98}-a_{97}x^{99}\]By comparing the coefficients on the left and the right, we can write the following system of many equations.\[\begin{align} C-a_0 &= -1 \\D-a_1 &=0 \\ a_0-a_2&=0 \\ &\vdots \\ a_{98}&=0 \\ a_{99}&= 1 \end{align}\]That gives \(C=-1\) and \(D=0\). Hmm.... I'd go with the simpler method above lol.

  7. Callisto
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Thanks!!

  8. yakeyglee
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Actually, kidding; it does give the same answer...I just was erronous in writing my system of equations... the last two subscripts should be 96 ad 97, which subsequently affect all of the previous equations in a chain which makes the appropriate values of \(C\) and \(D\).

  9. Callisto
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Hmm.. I'll stick to the first method here... since I haven't heard of the second method.. :(

  10. yakeyglee
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I made it up lmao. It works, though, in theory.

  11. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find 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
  • 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.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.