proof needed: a cubic equation has odd number of real roots if it has more than one


by lmamaths
Tags: cubic, equation, number, proof, real, roots
lmamaths
lmamaths is offline
#1
Apr22-05, 07:27 AM
P: 6
Hi,

A cubic equation has at least one real root.
If it has more than one why are there always an
odd number of real roots? Why not an even number
of real roots?

Can someone help me to prove this?

Thx!
LMA
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dextercioby
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#2
Apr22-05, 07:31 AM
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Because it has minimum one.In that case,if u factor that monom,u're left with a second order polynomial in "x" which has either 0 or 2 real roots...In total,the cubic has either 1 or 3 real roots.

Of course,the coefficients of the polynomials must be real.(in the other thread,too).

Daniel.
Galileo
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#3
Apr22-05, 03:23 PM
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If you allow complex numbers, you can prove that if you have a solution to a polynomial with real coefficients, its complex conjugate will also be a solution.
That means a polynomial of odd degree always has a real root. Moreover, if you have nonreal root, then you always have another one (its complex conjugate).

robert Ihnot
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#4
Apr23-05, 10:40 AM
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proof needed: a cubic equation has odd number of real roots if it has more than one


Galileo: you can prove....its complex conjugate will also be a solution.
This is because F(x) = Real part + imaginary part. So that for the function to go to zero, BOTH the real and imaginary parts go to zero, and so by changing the sign on the imaginary part of x, the complex conjugate will also go to zero.

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kleinwolf
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#5
Apr23-05, 03:31 PM
P: 309
Quote Quote by lmamaths
Hi,

A cubic equation has at least one real root.
If it has more than one why are there always an
odd number of real roots? Why not an even number
of real roots?

Can someone help me to prove this?

Thx!
LMA
If you play on the defintion of root, it can have an even number of real ones. For counterexample :

P(x)=(x-a)*(x-b)^2
jdavel
jdavel is offline
#6
Apr23-05, 05:18 PM
P: 618
Imamaths,

Graphing some functions of the form y = ax^3+bx^2+cx+d might help you to see what's going on.


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