find the eigen value for λ^3-11λ^2+40λ+36=0

To find the eigenvalues of the given cubic polynomial equation:

\[
\lambda^3 - 11\lambda^2 + 40\lambda + 36 = 0
\]

We can use the following approach:

1. **Identify possible rational roots:** 
   The Rational Root Theorem suggests that the possible rational roots are factors of the constant term (36) divided by factors of the leading coefficient (1). The possible rational roots are:

   \[
   \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36
   \]

2. **Test the roots:**
   Substitute each of these possible roots into the equation to see if any satisfy it (i.e., make the equation equal to 0).

Let's start by testing some of these roots.

Testing \( \lambda = 2 \):

\[
\lambda^3 - 11\lambda^2 + 40\lambda + 36 = 2^3 - 11(2^2) + 40(2) + 36 = 8 - 44 + 80 + 36 = 80 - 44 + 36 = 80 - 80 = 0
\]

Thus, \( \lambda = 2 \) is a root.

Now, we can factor the polynomial using \( \lambda - 2 \) as one factor. Let's perform synthetic division or factor the cubic polynomial to find the remaining factors.

Dividing \( \lambda^3 - 11\lambda^2 + 40\lambda + 36 \) by \( \lambda - 2 \):

The quotient will be \( \lambda^2 - 9\lambda - 18 \).

So, we have:

\[
\lambda^3 - 11\lambda^2 + 40\lambda + 36 = (\lambda - 2)(\lambda^2 - 9\lambda - 18)
\]

3. **Solve the quadratic equation:**

Now, solve the quadratic equation:

\[
\lambda^2 - 9\lambda - 18 = 0
\]

Using the quadratic formula:

\[
\lambda = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(-18)}}{2(1)} = \frac{9 \pm \sqrt{81 + 72}}{2} = \frac{9 \pm \sqrt{153}}{2}
\]

Since \( \sqrt{153} \) simplifies to \( 3\sqrt{17} \), the roots are:

\[
\lambda = \frac{9 \pm 3\sqrt{17}}{2}
\]

Thus, the eigenvalues of the equation are:

\[
\lambda_1 = 2, \quad \lambda_2 = \frac{9 + 3\sqrt{17}}{2}, \quad \lambda_3 = \frac{9 - 3\sqrt{17}}{2}
\]

Would you like further details or have any questions?

### Relative Questions:
1. How do we apply the Rational Root Theorem to find possible roots?
2. Can you explain the synthetic division process in more detail?
3. How is the quadratic formula derived and applied here?
4. What are the conditions for a cubic equation to have one real root and two complex roots?
5. How do you interpret the roots in terms of eigenvalues?

### Tip:
When solving polynomial equations, always check for rational roots first, as they can significantly simplify the process of factorization.

Learn how to find the eigenvalues of the cubic polynomial equation λ³ - 11λ² + 40λ + 36 = 0. This problem involves applying the Rational Root Theorem, using synthetic division, and solving a resulting quadratic equation. Suitable for advanced high school and undergraduate students.

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