Question

f(x)=x^4+2x^3-4x^2-x+1
let a,b,c,d be the roots of this
ab, ac, ad, bc, bd, cd are roots of polynomial g(x)
what is g(1) if g(0)=1

Answer 1

wow i am stumped, must be something to do with vieta's formulas, elementary symmetric functions
g(1) is the sum of the coefficients, so that is what you are being asked for, but i am not sure how to get them.
do you have any clues?

Answer 2

when you add these together you should get
\[\frac{a_2}{a_4}=\frac{-4}{1}=-4\]

Answer 3

and also
\[a+b+c+d=-\frac{a_3}{a_4}=-2\] if that helps

Answer 4

i will have to think on this, converting roots of one poly to roots of another, but if you get an answer please post, i would love to see it

Answer 5

I've been pondering with this the entire day. The problem asks us to find g(1).

Answer 6

this is what we know about a, b , c, d from vieta,

\[a+b+c+d=-2\]
\[ab+ac+ad+bc+bd+cd=-4\]
\[abc+abd+acd+bcd=1\]
\[abcd=1\]

Answer 7

we also know that we have a sixth degree polynomial that looks like
\[(x-ab)(x-ac)(x-ad)(x-bc)(x-bd)(x-cd)\]

Answer 8

and we are being asked for the sum of the coeffidents, because that is what
\[g(1)\] gives

Answer 9

Let a,b,c, and d be distinct roots of \[f(x)=x^4+2x^3-4x^2-x+1\]The six values ab,ac,ad,bc,bd and cd are roots of the polynomial \[g(x)\]
If \[g(0)=1\] find \[g(1)\]

Answer 10

If that helps

Answer 11

ok maybe we are getting somewhere

Answer 12

no i got the question

Answer 13

you are being asked for the sum of the coefficents of g

Answer 14

i think this might be hard, but i am not sure. where did the problem come from?

Answer 15

I was working on it with a teacher for fun the other day, neither of us found an answer, even as he was looking up theorems to help solve it

Answer 16

i think you need vieta's formula, the one that gives the coefficients of a polynomial in terms of the sum and the product of the roots. you can look here
http://en.wikipedia.org/wiki/Vieta%27s_formulas
or in this nice pdf attachment on page 68

Answer 17

keep in mind that g(1) is the sum of the coefficents, so that is what you are being asked for, and g(0)=1 tells you that the product of the roots
\[a^3b^3c^3d^3=1\]

Answer 18

Alright, I'll try to work off that for the rest of the night. Thank you very much!

Answer 19

i am afraid i was not much help
but it might be instructive to do a simple example, one with the roots that you know

Answer 20

say
\[x^4-14 x^3+71 x^2-154 x+120=(x-2)(x-3)(x-4)(x-5)\] then take the product of the roots taken two at a time and see how the coefficients relate

Answer 21

\[(x-6)(x-8)(x-10)(x-12)(x-15)(x-20)=x^6-71 x^5+2036 x^4-30196 x^3+244320 x^2-1022400 x+1728000\]

Answer 22

sum of coefficients is 921690

Answer 23

ok so much for that bright idea, i don't see the connection, but who knows. i will post also so maybe tomorrow someone will give a good explanation

Answer 24

Thank you very much