
#1
Oct909, 09:07 PM

P: 186

1. The problem statement, all variables and given/known data
Prove that for any polynomial function f, and any number a, there is a polynomial function g, and a number b, such that f(x) = (xa)g(x) + b for all x. 2. Relevant equations 3. The attempt at a solution Base Case: Let f(x) be a polynomial of degree n = 1. Then: [tex] \begin{align*} f(x) &= c_1x+c_0\\ f(x) &= c_1x  c_1a + c_1a + c_0\\ f(x) &= (xa)c_1 + c_1a + c_0\\ f(x) &= (xa)g(x) + b \end{align*} [/tex] where g(x) = c_{1} and b = (c_{1}a + c_{0}). Let P(n) be the statement that every polynomial function of degree n can be written as f(x) = (xa)g(x) + b. Take P(n+1) and assume P(n) is true: [tex] \begin{align*} f(x) &= c_{n+1}x^{n+1} + c_{n}x^{n}+...+c_1x + c_0\\ f(x) &= c_{n+1}x^{n+1} + (xa)q(x) + k_0\\ f(x) &= c_{n+1}x^{n+1}  c_{n+1}x^na + c_{n+1}x^na + (xa)q(x) + k_0\\ f(x) &= (xa)(c_{n+1}x^n) + c_{n+1}x^na + (xa)q(x) + k_0\\ f(x) &= (xa)(c_{n+1}x^n) + (xa)r(x) + k_1 + (xa)q(x) + k_0\\ f(x) &= (xa)(c_{n+1}x^n + r(x) + q(x)) + k_1 + k_0\\ f(x) &= (xa)g(x) + b \end{align*} [/tex] where g(x) = c_{n+1}x^{n} + r(x) + q(x) and b = k_{1} + k_{0}. I think it's right, but I'm always so unsure of myself when it comes to induction. 



#2
Oct909, 10:15 PM

Sci Advisor
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P: 25,167

That seems ok to me. It might help a bit if you explicitly say why you can make certain substitutions because of the induction hypothesis. But I think it's basically ok. And as a technical point you can't just prove P(n+1) from P(n). c_n might be 0. P(n) should really be "every polynomial function of degree k<=n can be written as f(x) = (xa)g(x) + b". But that's really just a legal fine point.




#3
Oct909, 11:34 PM

P: 186

thanks very much.




#4
Dec2209, 01:40 AM

P: 148

Does my proof make sense?
Hi,
could someone explain how we go from 1. to 2. in the expressions below? I fail to see how [tex]c_{n+1}x^na=(xa)r(x)+k_1[/tex] Thanks. [tex] \begin{align*} 1.f(x) &= (xa)(c_{n+1}x^n) + c_{n+1}x^na + (xa)q(x) + k_0\\ 2.f(x) &= (xa)(c_{n+1}x^n) + (xa)r(x) + k_1 + (xa)q(x) + k_0\\ \end{align*} [/tex] 



#5
Dec2209, 02:14 AM

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P: 25,167

It's the induction hypothesis. You assume every polynomial of degree <=n can be written as (xa)*r(x)+k. Think about it. The OP didn't state this as clearly as should have been done. See if you can do better. Look at the n=1 case. If you understand that, you understand everything. Also think about posting your own threads.




#6
Dec2209, 03:24 AM

P: 148

if [tex](xa)[/tex] is a factor of [tex]f(x)[/tex] then [tex]f(a)=0[/tex]. [tex]f(\frac{c_0}{c_1})=c_1(\frac{c_o}{c_1})+c_0 =0 [/tex] The Remainder Theorem gives that: [tex]f(a)=b[/tex], so [tex]f(x)=(x(\frac{c_o}{c_1}))c_1+c_1(\frac{c_o}{c_1})+c_0[/tex] I guess this is all good and well if [tex](xa)[/tex] is a factor of [tex]f(x)[/tex]. But if it isn't then, [tex]f(x)=(xa)c_1 + c_1(a)+c_0[/tex] I've successfully managed to confuse myself. As for making my own threads, I always read how one should do a proper search before posting. I thought it would be better not to make my own thread about something that has been covered in the past. Will do so in the future though, thanks. 



#7
Dec2209, 09:13 AM

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P: 25,167

n=1. f(x)=c1*x+c0=c1*xc1*a+c1*a+c0=c1*(xa)+(c1*a+c0). c1*a+c0 is a number. c1 is a 'polynomial' of degree less than 1. Now lets do n=2. f(x)=c2*x^2+c1*x+c0=c2*x^2c2*x*a+c2*x*a+c1*x+c0=c2*x*(xa)+(c2*x*a+c1*x+c0). Since (c2*x*a+c1*x+c0) is a polynomial of degree one, by the induction hypothesis (and as we showed above) it can be written as p(x)(xa)+b where p(x) has degree <1 (e.g. it's really just a constant). So f(x)=c2*x*(xa)+p(x)(xa)+b=(c2*x+p(x))*(xa)+b. So f(x)=q(x)*(xa)+b where q(x)=c2*x+p(x) has degree 1. Try doing n=3 explicitly.
Yep, researching is a great idea. In preparation for solving the problem or posting your own thread. I don't necessarily even pay much attention to add ons to ancient posts unless there are my own. If I see some other post with 5+ replies to it, I'll figure the problem is already being dealt with and move on. You'll get more replies faster if it's a new thread. 



#8
Dec2209, 11:16 PM

P: 148

Hi,
Here's my try at the n=3 case: [tex] \begin{tabular}{ r c l } \(f(x)\) & \(=\) & c_3x^3+c_2x^2+c_1x+c_0\) \\ & \(=\) & c_3x^3c_3x^2a+c_3x^2a+c_2x^2+c_1x+c_0\) \\ & \(=\) & (xa)c_3x^2+c_3x^2a+c_2x^2+c_1x+c_0\) \end{tabular} [/tex] [tex]c_3x^2a+c_2x^2+c_1x+c_0[/tex] is a polynomial of degree two, and can, by the induction hypothesis be written as [tex](xa)p(x)+b[/tex]. [tex] \begin{tabular}{ r c l } \(f(x)\) & \(=\) & (xa)c_3x^2+(xa)p(x)+b\) \\ & \(=\) & (xa)\left(c_3x^2+p(x)\right)+b\) \\ \end{tabular} [/tex] [tex]q(x)=c_3x^2+p(x)[/tex] has degree 2, and [tex]b[/tex] has degree 1. Thank you for being patient! 



#9
Dec2309, 09:17 AM

Sci Advisor
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P: 25,167

That's nice. It was worth waiting for.




#10
Dec2309, 10:50 PM

P: 148

You are too kind, thanks. I feel comfortable with this now.



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