Discriminant of higher order polynomials.
I was just slightly curious as to how the formulas for the discriminants of higher order polynomials are obtained.
I know the discriminant of a quadratic is and just wondered where we got this result from and how we would go about generalizing it to higher order equations (from what I've seen from the determinant of a cubic, it seems to get messy very quickly!
)
Also, what meaning would it have in the context of higher order equations? Would it just tell you the same thing as the discriminant t of a quadratic?
Posted from TSR Mobile
I know the discriminant of a quadratic is and just wondered where we got this result from and how we would go about generalizing it to higher order equations (from what I've seen from the determinant of a cubic, it seems to get messy very quickly!
) Also, what meaning would it have in the context of higher order equations? Would it just tell you the same thing as the discriminant t of a quadratic?
Posted from TSR Mobile
(edited 9 years ago)
I think you mean discriminant. For a general polynomial with real coefficients we can say:
If it equals 0 you have at least one repeated root.
If it's < 0 you have an odd number of pairs of complex conjugate roots. (I.e. 2(2k+1) complex roots for some integer k).
If > 0 you have an even number of pairs of complex conjugate roots.
If it equals 0 you have at least one repeated root.
If it's < 0 you have an odd number of pairs of complex conjugate roots. (I.e. 2(2k+1) complex roots for some integer k).
If > 0 you have an even number of pairs of complex conjugate roots.
Original post by DFranklin
I think you mean discriminant. For a general polynomial with real coefficients we can say:
If it equals 0 you have at least one repeated root.
If it's < 0 you have an odd number of pairs of complex conjugate roots. (I.e. 2(2k+1) complex roots for some integer k).
If > 0 you have an even number of pairs of complex conjugate roots.
If it equals 0 you have at least one repeated root.
If it's < 0 you have an odd number of pairs of complex conjugate roots. (I.e. 2(2k+1) complex roots for some integer k).
If > 0 you have an even number of pairs of complex conjugate roots.
Yes, that was it, doing too much work on matrices gets confusing
Right, thanks for helping
How would you go about calculating the expression for this discriminant for a general polynomial?Posted from TSR Mobile
The wiki page for the discriminant explains this:
http://en.wikipedia.org/wiki/Discriminant#Discriminant_of_a_polynomial
It's somewhat heavy going if you're only used to A-level maths.
http://en.wikipedia.org/wiki/Discriminant#Discriminant_of_a_polynomial
It's somewhat heavy going if you're only used to A-level maths.
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