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f(x) be a polynomial of degree atmost 7 which leaves remainder -1 and 1 on division by \((x-1)^4\) and \((x+1)^4\) respectively. If the sum of pairwise product of all roots of \(f(x)=0\) is n then the value of (5n + 24) is

 

P.S. final answer is 0, but can someone elaborate the solution? 

Thank you! :) 

 Jul 28, 2021
 #1
avatar+114511 
+2

I actually do not understand what this means

 

"If the sum of pairwise product of all roots of  f(x)=0  is n"    Is it even meant to be real like that??

 

I thought it meant that if you find the product of every pair of roots and add them together then the answer is 0. 

But that would mean that n has no meaning in the problem. 

So I am really confused.

 

Amy, do you understand what the question is asking?  If so please explain it to me.   indecision

 Aug 1, 2021
 #2
avatar+524 
+2

Melody,

 

From what I understand it says that f(x) is a polynomial of degree atmost 7 right? then it would have atmost 7 roots.

Let's then consider its roots are a,b,c,d,e,f,g then, sum of pairwise product of all roots means ab + bc + cd + de + ef + fg + ag + bg + cg +...+ eg + af + bf +.... so on = n you get the point right?

I think then we would've to find that sum to find the value of n. 

 

Another thing, its also given that f(x) leaves remainder -1 and 1 on division with (x-1)4 and (x+1)4 I don't know what to deduce from that...

amygdaleon305  Aug 1, 2021
 #3
avatar+2207 
+2

Woah, this is a hard problem, I can't figure it out, but maybe this is helpful. 

 

f(x) = a(x-1)^4 - 1, where a is at most 3 degrees. 

f(x) = b(x+1)^4 + 1, where b is at most 3 degrees. 

This would mean that f(1) = -1, f(-1) = 1. 

The sum of the coeficients is -1, and the even degree coefficients - odd degree coefficients = 1. 

By vieta's, ax^(n) + bx^(n-1) + cx^(n-2)..., the pairwise product of all the roots would be c/a. 

Good luck. :))

 

=^._.^=

 Aug 1, 2021
 #4
avatar+524 
+2

Yeah that's very good but then in order to find the value of n we would've to find the value of c/a which are the coefficients in f(x). 

Anyway, thanks a lot catmg! :D Let me see what can be done further

amygdaleon305  Aug 1, 2021
 #5
avatar+524 
+1

Has anyone had any luck with this question so far? I tried everything I could, but it only seem to get more confusing frown

 Aug 2, 2021
 #6
avatar+2207 
+1

I can't seem to figure it out. :((

 

=^._.^=

catmg  Aug 3, 2021
 #7
avatar+524 
+1

Its okay, its a really hard question... it may take some time to get solved 

amygdaleon305  Aug 4, 2021

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