Eigenvalue/Eigenvector

I am loosing my mind. I cannot seem to factor the http://www.physlink.com/Images/symbols/lambda.gif^3-4http://www.physlink.com/Images/symbols/lambda.gif^2-10http://www.physlink.com/Images/symbols/lambda.gif-12.

My problem is to verify that is an eigenvector of A=(1,2,3), (3,2,1), (2,3,1).

det(http://www.physlink.com/Images/symbols/lambda.gif-AI) = (http://www.physlink.com/Images/symbols/lambda.gif-1)(http://www.physlink.com/Images/symbols/lambda.gif-2)(http://www.physlink.com/Images/symbols/lambda.gif-1)+(-2)(-1)(-2)+(-3)(-3)(-3)-(-2)(http://www.physlink.com/Images/symbols/lambda.gif-2)(-3)-(-1)(-3)(http://www.physlink.com/Images/symbols/lambda.gif-1)-(http://www.physlink.com/Images/symbols/lambda.gif-1)(-2)(-3)
= http://www.physlink.com/Images/symbols/lambda.gif^3-4http://www.physlink.com/Images/symbols/lambda.gif^2-10http://www.physlink.com/Images/symbols/lambda.gif-12
= ???

since you are given the eigenvector, just calculate:



this tells you that 6 is an eigenvalue.

now divide the characteristic polynomial by λ-6.