Find Eigen Values: Solve Char. Equation |A-#I| = 0 of Degree 3

[zeroxff0000ff]

hello everybody,

Consider '#' as lamda.
How to find roots(eigen values) of characteristic equation |A-#I| = 0.
I know how to find it it using numerical methods.
But can anyone please show me how to procced for degree 3 equations.

thanks and regards.

 

[HallsofIvy]

Solve the equation! If you cannot factor it and don't want to use a numerical approximation, you could use Cardano's cubic formula:
http://www.math.vanderbilt.edu/~schectex/courses/cubic/

 

[zeroxff0000ff]

thanks HallsofIvy.

just now i found some methods like factor and remainder theorem from the following link :
http://www.themathpage.com/aprecalc/factor-theorem.htm

can you please tell me advantage and disadvantage of using factor and reminder theorem.

 

[Office_Shredder]

I'm not sure what the remainder theorem is, but the factor theorem is usually what's used to solve cubics... if there's an easy root (like 1, 2, 0, -1) you can reduce the degree of your equation to a quadratic fairly easily, and from there solve for the remaining roots.

You can use the integer root theorem (on the website you posted) to figure out which integers are worth guessing. The advantage is that most problems will have a root that you can find in this way. The disadvantage is that if the polynomial doesn't have an integer root, you can't find any roots using this method. But it doesn't take a lot of time to try, so usually this is the best way to start.

 

[mathwonk]

the remainder theorem is the theorem from high school algebra that says the remainder of f(x) after dividing it by x-a, is f(a).

 

[HallsofIvy]

the remainder theorem is the theorem from high school algebra that says the remainder of f(x) after dividing it by x-a, is f(a).

And in particular, if f(a)= 0, the remainder is 0 so x-a is a factor of f(x).

It seems very strange to me that a person would be working with eigenvalues while not knowing how to solve polynomial equations by factoring!