Roots of Cubic Equations

Date: 01/13/2002 at 00:03:55
From: John Devlin
Subject: Roots of Cubic Equations

Can you help me with this?

a, b, and c are the roots of the equation x^3 - px^2 + qx - r = 0.

Express the following in terms of p, q, and r.

   1) a^2 + b^2 + c^2

   2) 1/a^2 + 1/b^2 + 1/c^2

Thank you very much.

Date: 01/15/2002 at 17:04:28
From: Doctor Schwa
Subject: Re: Roots of Cubic Equations

>1) a^2 + b^2 + c^2

Hint: a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab + ac + bc).

Does that help?

>2) 1/a^2 + 1/b^2 + 1/c^2

Hint: If a is a root of the given polynomial, then 1/a is a root of 
(1/x)^3 - p(1/x)^2 + q(1/x) - r, and then multiplying by x^3, 1/a is 
also a root of -rx^3 + qx^2 -px + 1, and then dividing through by -r 
will put it in the same pattern as your problem (1).

I hope that helps. These are neat problems, the kind I'd expect to see 
on a math contest more than in the typical math class.

Feel free to write back if you'd like another hint.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/