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

We need to find the eigenvalues of the cubic equation:

\[
λ^3 - 11λ^2 + 40λ - 36 = 0
\]

To do this, we will attempt to factor the cubic polynomial by checking possible rational roots (based on the Rational Root Theorem) and then solve for \( λ \).

### Rational Root Theorem
The Rational Root Theorem suggests that any possible rational root of a polynomial equation with integer coefficients is of the form \( \frac{p}{q} \), where:
- \( p \) is a factor of the constant term (here \( -36 \)),
- \( q \) is a factor of the leading coefficient (here \( 1 \)).

Thus, the possible rational roots are the factors of \( -36 \):  
±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36.

Let's test these possible roots using synthetic division or direct substitution. I'll check them step by step.import sympy as sp

# Define the variable and polynomial
λ = sp.symbols('λ')
polynomial = λ**3 - 11*λ**2 + 40*λ - 36

# Find the possible rational roots by evaluating the polynomial at integer values
possible_roots = [-36, -18, -12, -9, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 9, 12, 18, 36]

# Check which of these values are roots of the polynomial
roots = [r for r in possible_roots if polynomial.subs(λ, r) == 0]
rootsIt appears that none of the possible rational roots from the Rational Root Theorem are actual roots of the polynomial. Since the polynomial doesn't have rational roots, we'll proceed by finding the roots numerically or using symbolic methods to factor the cubic equation and determine the eigenvalues.

Let me calculate the exact eigenvalues using symbolic methods.# Solve the cubic equation symbolically for the eigenvalues
eigenvalues = sp.solve(polynomial, λ)
eigenvaluesThe exact symbolic roots of the cubic equation \( λ^3 - 11λ^2 + 40λ - 36 = 0 \) are quite complex, involving cubic roots and imaginary components. To provide a clearer understanding, let's approximate these roots numerically.# Get the numerical approximations of the eigenvalues
numerical_eigenvalues = [sp.N(root) for root in eigenvalues]
numerical_eigenvaluesThe eigenvalues of the cubic equation \( λ^3 - 11λ^2 + 40λ - 36 = 0 \) are approximately:

1. \( 4.84 + 1.94i \)
2. \( 4.84 - 1.94i \)
3. \( 1.32 \)

These are a pair of complex conjugate roots and a single real root.

Would you like more details on the solving process or have any further questions?

Here are some related questions you might find helpful:
1. What are eigenvalues and how are they used in linear algebra?
2. How can we solve cubic equations using factorization?
3. What is the Rational Root Theorem, and how does it help in finding roots?
4. What are the properties of complex roots in polynomials?
5. How can we graph cubic equations to visualize their roots?

**Tip:** For cubic equations, if rational roots are hard to find, using numerical methods often helps to approximate the solutions efficiently.

Learn how to find the eigenvalues of the cubic equation λ^3 - 11λ^2 + 40λ - 36 = 0 using symbolic and numerical methods. Explore the application of complex numbers in solving cubic equations. Suitable for college-level mathematics.

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