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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 ofii) Show that
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 rootsThank you very much in advance!
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..."
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