Proving Odd Number of Real Roots in Cubic Equations

• lmamaths
In summary, the conversation discusses the concept of real roots in cubic equations. It is stated that a cubic equation must have at least one real root, and if it has more than one, there will always be an odd number of real roots. This is because for a cubic equation to have multiple real roots, it must have either 0 or 2 real roots after factoring out a monomial. The conversation also touches on the idea that allowing complex numbers can result in an even number of real roots. Additionally, the conversation mentions Einstein's disdain for mathematics and his focus on physics instead.
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

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.

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).

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.

Einstein, before fleeing Germany, had already become a refugee from mathematics. He later said that he could not find, in that garden of many paths, the one to what was fundamental. He turned to the more earthly domain of physics, where the way to the essential was, he thought, clearer. His disdain for mathematics earned him the nickname "lazy dog" from his teacher, Hermann Minkowski, who would soon recast the "lazy dog's" special relativity into its characteristic four-dimensional form. "You know, once you start calculating," Einstein would quip, "you **** yourself up before you know it." http://chronicle.com/temp/reprint.php?%20id=7ixqqc97xiroy9hnb9o2154f61c2wl02

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lmamaths said:
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

Imamaths,

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

1. How do you determine the number of real roots in a cubic equation?

To determine the number of real roots in a cubic equation, you can use the discriminant method. This involves calculating the discriminant of the equation, which is b^2-4ac, where a, b, and c are the coefficients of the cubic equation. If the discriminant is positive, there are three distinct real roots. If it is zero, there is one real root. If it is negative, there are no real roots, but there may be complex roots.

2. Why is it important to prove the number of real roots in a cubic equation?

Knowing the number of real roots in a cubic equation is important because it helps us understand the behavior of the equation and its solutions. It also allows us to determine if the equation can be solved using real numbers or if we need to use complex numbers.

3. What is the relationship between the degree of a polynomial and the number of its roots?

The degree of a polynomial is the highest power of the variable in the equation. For a cubic equation, the degree is 3. The fundamental theorem of algebra states that a polynomial of degree n has exactly n complex roots. Therefore, a cubic equation can have up to 3 complex roots, which may be a combination of real and imaginary roots.

4. Can a cubic equation have an odd number of real roots?

Yes, a cubic equation can have an odd number of real roots. This occurs when the discriminant is positive and there are three distinct real roots, or when the discriminant is zero and there is one real root. In both cases, the number of real roots is odd.

5. Are there any shortcuts or tricks to proving the number of real roots in a cubic equation?

Yes, there are some shortcuts and tricks that can help you quickly determine the number of real roots in a cubic equation. For example, if the sum of the coefficients of the equation is zero, then the equation has at least one real root. Additionally, if the equation has a linear or quadratic factor with all real coefficients, then the remaining factor must have real roots. However, these shortcuts may not always apply and it is important to use the discriminant method for a more accurate determination of the number of real roots.

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