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Callisto
 3 years ago
Formulas and Identities #1
Question 31
Consider the identity \(x^{99}1 ≡ (x^21)P(x) + Cx+D\), where P(x) is a polynomial in x. Find the values of C and D
Callisto
 3 years ago
Formulas and Identities #1 Question 31 Consider the identity \(x^{99}1 ≡ (x^21)P(x) + Cx+D\), where P(x) is a polynomial in x. Find the values of C and D

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Callisto
 3 years ago
Best ResponseYou've already chosen the best response.1Bingo ... How did you get it?!?

dpaInc
 3 years ago
Best ResponseYou've already chosen the best response.2silly me... i wrote the proof of it on the margin of my notebook and now i can't find it....:(

dpaInc
 3 years ago
Best ResponseYou've already chosen the best response.2substitute 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...

yakeyglee
 3 years ago
Best ResponseYou've already chosen the best response.0Well, 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 \(x^21\) gives us the following.\[(x^21)P(x) = x^2P(x)P(x) =\]\[= a_0  a_1 x + (a_0  a_2)x^2 + \cdots (a_{95}a_{97})x^{97}a_{96}x^{98}a_{97}x^{99} \]Now, let's add \(Cx+D\).\[x^{99}1=(Ca_0) + (Da_1) x + (a_0  a_2)x^2 + \cdots (a_{95}a_{97})x^{97}a_{96}x^{98}a_{97}x^{99}\]By comparing the coefficients on the left and the right, we can write the following system of many equations.\[\begin{align} Ca_0 &= 1 \\Da_1 &=0 \\ a_0a_2&=0 \\ &\vdots \\ a_{98}&=0 \\ a_{99}&= 1 \end{align}\]That gives \(C=1\) and \(D=0\). Hmm.... I'd go with the simpler method above lol.

yakeyglee
 3 years ago
Best ResponseYou've already chosen the best response.0Actually, 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 \(D\).

Callisto
 3 years ago
Best ResponseYou've already chosen the best response.1Hmm.. I'll stick to the first method here... since I haven't heard of the second method.. :(

yakeyglee
 3 years ago
Best ResponseYou've already chosen the best response.0I made it up lmao. It works, though, in theory.
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