λ^3-2λ^2-5λ+6=0 solve to get eigen values

To solve the cubic equation \( \lambda^3 - 2\lambda^2 - 5\lambda + 6 = 0 \) for its roots (which are the eigenvalues), we can follow these steps:

### Step 1: Factorization by trial and error (Rational Root Theorem)
We'll check for possible rational roots using the Rational Root Theorem, which states that any possible rational root is a factor of the constant term (6) divided by a factor of the leading coefficient (1). 

The possible rational roots are:
\[ \pm 1, \pm 2, \pm 3, \pm 6 \]

We can test these roots by substitution:

1. **Test \(\lambda = 1\):**
   \[
   1^3 - 2(1)^2 - 5(1) + 6 = 1 - 2 - 5 + 6 = 0 
   \]
   So, \(\lambda = 1\) is a root.

### Step 2: Polynomial division
Since \(\lambda = 1\) is a root, we can factor the cubic polynomial as:
\[
\lambda^3 - 2\lambda^2 - 5\lambda + 6 = (\lambda - 1)(\lambda^2 - \lambda - 6)
\]

### Step 3: Factor the quadratic polynomial
Now, we need to factor the quadratic \( \lambda^2 - \lambda - 6 \).

We can factor it as:
\[
\lambda^2 - \lambda - 6 = (\lambda - 3)(\lambda + 2)
\]

### Step 4: Find the roots
Now, we have the factorized form of the cubic equation:
\[
(\lambda - 1)(\lambda - 3)(\lambda + 2) = 0
\]

The roots (eigenvalues) are:
\[
\lambda_1 = 1, \quad \lambda_2 = 3, \quad \lambda_3 = -2
\]

So, the eigenvalues are \( \lambda = 1, 3, -2 \).

Would you like further details on any step? 

### Related Questions:
1. How do you apply the Rational Root Theorem?
2. What are the properties of eigenvalues for a given matrix?
3. Can eigenvalues be complex numbers?
4. How do you find eigenvectors corresponding to these eigenvalues?
5. What is the significance of the determinant in finding eigenvalues?

### Tip:
When solving cubic equations, always check for simple rational roots first—they can significantly simplify the process.

Learn how to solve the cubic equation λ^3 - 2λ^2 - 5λ + 6 = 0 to find its eigenvalues. This problem involves applying the Rational Root Theorem and factorization techniques to identify λ values satisfying the equation. Suitable for students in Grades 10-12.

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