Polynomial questions (1 Viewer)

[Constip8edSkunk]

Can someone help me on 2 questions, both from cambridge (further questions 4: q11 and q12) :

1/ the eqation x^n+px-q=0 has a double root. Show that
(p/n)^n+[q/(n-1)]^(n-1)=0


2/ The roots of the equation n(Sigma)r=1 [ arx^(n-r) ]= 0 , where a0 =1, are the the first n positive intergers. Show that

a2 = (n+1)(n-1)(3n+2)n/24


Thanx

 

[maniacguy]

1/ the eqation x^n+px-q=0 has a double root. Show that
(p/n)^n+[q/(n-1)]^(n-1)=0

Let the double root be at x = a

Differentiate:
nx^(n-1) + p = 0 also has a root at the point x = a.

Then a^(n-1) = -p/n

Now, a^n + pa - q = 0
a(a^(n-1) + p) - q = 0
a(-p/n + p) - q = 0
ap(n-1)/n = q

ap/n = q/(n-1)

a^(n-1)*(p/n)^(n-1) = (q/(n-1))^(n-1)

The result follows...

For the second question:

Clearly the polynomial is (x-1)(x-2)...(x-n) as those are the stated roots and we are told it is monic.

Hence a2 = sum of products in pairs = 1*2 + 1*3 + ... + n*4 + ...
Note this is simply [(1+...+n)^2 - (1^2 + .... + n^2)]/2
= [[n(n+1)/2]^2 - n(n+1)(2n+1)/6]/2
= [n(n+1)/2 * [n(n+1)/2 - (2n+1)/3]]/2
= [n(n+1)/2 * [3n^2 + 3n - 4n - 2]/6]/2
= [n(n+1)/12 * [3n^2 -n - 2]]/2
= [n(n+1)/12 * (3n+2)(n-1)]/2
= n(n+1)(n-1)(3n+2)/24

 

[Constip8edSkunk]

Thanks alot!!! Ive been stumped for a few days already...

Theres one thing i dont quite get:

Why does 1^2 + .... + n^2 = n(n+1)(2n+1)/6?

again thanx for finding the time to help

 

[McLake]

Originally posted by Constip8edSkunk
Theres one thing i dont quite get:

Why does 1^2 + .... + n^2 = n(n+1)(2n+1)/6?

Beause it does! There is probably some rule for summing a series like this (which isn't an AP or a GP) and thats what the formula looks like.

 

[Constip8edSkunk]

ah. we havent learnt it yet then . thanks for your help.

 

[maniacguy]

it's a fairly simple proof by induction (in the harder 3u section of the 4u course)

you should do it when you do induction...

 

[SBHS4EVER]

CONSTIPATED SKUNK

Who is this- i know u go to SBHS since u have the same name on our Year 12 website

 

[Constip8edSkunk]

Hmmm, done induction but i never remember having to memorise that formula

SBHS4EVER: john... my names on the memberlist on that website anyway....

Edit: you are paul rite?

 

[turtle_2468]

Some questions (particularly harder ones) in cambridge require you to know some miscellaneous facts. The two most useful formulas in induction you should probably memorise, as you use them a fair bit (in my experience anyway)...
the 1+2+...+n one
and the squares one mentioned above.

 

[Constip8edSkunk]

ok thanx for the advice.