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ajprincess
Please help:) Find the relation between \(\alpha\), \(\beta\), \(\gamma\) in the order that \(\alpha+\beta x+\gamma x^2\) may be expressible in one term in the factorial notation.
maybe you can google a similiar question, and i will reply
ok this is not what i am use to, algebra. what are you studying? what book
engineering mathematics by amit k awasthi
hmm no. I am nt studying any particular book. this question was given to me by my lecturer.
what is a factorial polynomial, can you give me a definition
A factorial polynomial \(x^p\) is defined as \(x^p=x(x-h)(x-2h)--------------(x-(p-1)h)\) where p is a positive integer.
so, here take p= 2, a+bx+cx^2=x(x-h) and find relation between a,b,c.
p=2 to make factorial polynomial as quadratic
jst a sec. am workng on it
I am nt sure if what I have done is right. a=0, b=-h and c=1. am I right @hartnn?
but that doesn't give u relation between them.......
ya i am greatly confused
When I googled my question i found this link. http://acadmedia.wku.edu/Zhuhadar/eBooks/0977858251-STATISTICAL.pdf page number 229. i am trying to understand it
Suppose \(\alpha+\beta x+\gamma x^2=(u+vx)^2\). This means \(u^2+2uvx+v^2x^2=\alpha+\beta x+\gamma x^2.\) Equating coefficients, we have that \(u^2=\alpha\), \(2uv=\beta\), and \(v^2=\gamma\). These three equations imply \(2\sqrt{\alpha\gamma}=\beta\).
OH. You're doing Knuth-esque math. Okay, one second.
The above answer is technically true, but it's not what the professor is looking for. I'll explain what he's doing in a moment . . .
It would appear your professor made a significant error. He's using what's called the rising factorial and incorrectly at that. (He or she, whoever.) The following is the definition of the notation \(a^{(n)}\): \[ a^{(n)}=a(a+1)(a+2)\cdots(a+n-1)=\prod_{1\le i\le n}\left(a+i-1\right).\] What your professor wants, I believe, is the following: Let \(\alpha+\beta x+\gamma x^2=(u+vx)^{(2)}\). Then, following the definition of the rising factorial, we have \[(u+vx)^{(2)}=\prod_{1 \le i \le 2}\left(u+vx+i-1\right)=(u+vx)(u+vx+1).\] Expanding that our, we get \[(u+vx)(u+vx+1)=u^2+uvx+u+uvx+v^2x^2+vx=u^2+u+2uvx+vx+v^2x^2.\] Since we have let \(\alpha+\beta x+\gamma x^2=(u+vx)^{(2)}\), we have that \(\alpha+\beta x+\gamma x^2=(u^2+u)+(2uv+v)x+v^2x^2.\) From this we can conclude (by "equating coefficients") \(\alpha=u^2+u\), \(\beta=2uv+v\), and \(\gamma=v^2\). To search for a relation between our three variables, substitue \(\pm \sqrt{\gamma}\) for \(v\): \(\beta=\pm2u\sqrt{\gamma}\pm\sqrt{\gamma}\). Solving this equation for \(u\), we see that \(u=\frac{\beta \pm \gamma}{\pm 2\sqrt{\gamma}}.\) Substitution of this into \(\alpha=u^2+u\) reveals \[\alpha=\frac{(\beta\pm \gamma)^2}{4\gamma}+\frac{\beta \pm \gamma}{\pm 2\sqrt{\gamma}},\] giving us the desired: a relation between \(\alpha, \beta\) and \(\gamma\).
Thank you soooo much for the explanation:)
is that way wrong? So which method should I use? Actually the answer I posted is not the answer my professor gave me. Actually she didnt give us any answer yet
I don't think the way was entirely wrong. It's just that the author misunderstood the definition of \(a^{(n)}\). You know what I mean?
When the writer of that post said \[(a+bx)^{(2)}=(a+bx)[a+b(x-1)],\] they were wrong and this messed up the entire problem. However, what they were _trying_ to do was right.
Does that make things more clear?
ya it is. thank u soooo much:)
You're welcome. But, I'd like to thank you for the interesting problem!
Let me go ahead and recommend--if you're getting problems like this a lot--Donald E. Knuth's Concrete Mathematics. In there, there's all kinds of this craziness: falling factorials, rising factorials, ceiling functions, floor functions, summations, discrete calculus, etc. It seems to directly pertain to what you're doing, but I can't be certain.
You're welcome! Have a wonderful weekend.
Thanks nd wish u the same:)