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thedarktiger

Member

Registered: 2014-01-10

Posts: 91

More of Vieta

These are driving me mad:

(1)Suppose the polynomial

has integer coefficients, and its roots are distinct integers.

Given that a_n=2 and a_0=66, what is the least possible value of

?

(2)Two of the roots of the polynomial

have product -32.

Determine k.

(3)Let

be the roots of f(x).

Compute

Last edited by thedarktiger (2014-12-12 21:07:00)

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Good. You can read.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: More of Vieta

hi thedarktiger,

Q1.  Odd ?  By Vieta, the sum of the roots will be -a sub(n-1)/asub(n)

So how small can the absolute value of the sum of the roots be ?

If n is odd, they could be 0,  ±p,  ±q,  ±r ......  In this case the sum of the roots is zero.  So I claim the minimum value is zero.  [Somehow, I don't think that can be right, but I cannot see why not.

EDIT:  Just had a fresh thought about this.  Working on it................................

That doesn't work because the product of the roots would be zero not 66/2 = 33

As the root are integers try a =  ±3 b =  ±11 and look for the smallest sum ('absoluted').

Q2.  Let the roots be a, b, c and d, and say ab = -32.

Then cd = -1984/-32 = +62

k = ab + ac + ad + bc + bd + cd = -32 + 62 + (a+b)(c+d)

Let P = a+b  and Q = c+d.  Then we have

k = 30 + PQ    ..............................(1)

Also

P + Q = 18    .................................(2)

and

-32c + 62b + 62a -32d = -200 = -32Q + 62P    ...................(3)

Solve (2) and (3) and substitute into (1)

I haven't looked at Q3 yet.

Bob

Last edited by Bob (2014-12-13 02:31:57)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: More of Vieta

Q3.

Should be straight forward to sum those.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: More of Vieta

The answer for q1 was 14. Interesting.
That means the poly was 2x^3+14x^2-82x+66.

q2 was 86 then.
Thank you for your help.

---

Good. You can read.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: More of Vieta

hi thedarktiger,

Q2.
I get 86 as well

Q1

So choose p = 11 and q =+3

Have you an answer for Q3 yet ?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob