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SaFrauya

Guest

Roots of Polynomial Equations: Proof Question and Misc Question

Hello, I'm a little stuck on this question...

The cubic equation

has roots

.

i) Write down the values of

and

, and express k in terms of

ii) Show that

SaFrauya

Guest

Re: Roots of Polynomial Equations: Proof Question and Misc Question

Sorry, that question was all wrong. >_<

It should be like this:

1) The cubic equation

has roots

.

i) Express p, q and r in terms of k and

.

ii) Show that

iii) Solve the equation for the case where p=q=-3

And the second question is this:

2) The equation

has roots

. Find a cubic equation with integer coefficients which has roots

Thank you very much in advance!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Roots of Polynomial Equations: Proof Question and Misc Question

1. This is how symmetric polynomials were first studied.  If you first assume the polynomial has three distinct roots, a, b and c, then you know that:

2x^3 + px^2 + qx + r = (x - a)(x - b)(x - c)

Now take the right side, and multiply everything out.  You may now equate coefficients (if you want a rigorous proof of this, it would be because x^n is linearly independent with x^m when viewed as being R-module for n not equal to m).

The rest of 1 should follow from this expression, but I haven't worked it out.  Just post back if you get lost.

2. Same idea, now you're looking at the polynomial:

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Roots of Polynomial Equations: Proof Question and Misc Question

1(iii) is very straightforward. You know that

. If

then they cancel out in the equation in 1(ii), leaving you with

. Use that to solve for

; once you have one root, the other two should be easily found.