x6 - 132x5 +7260x4 - 212960x3 + 3513840x2 - 30921792x + 113379904 = 0

The given equation is:

\[
x^6 - 132x^5 + 7260x^4 - 212960x^3 + 3513840x^2 - 30921792x + 113379904 = 0
\]

This is a degree-6 polynomial, and solving such equations algebraically can be quite complex. However, numerical methods or factoring (if applicable) are commonly used. I will first try to factor it or look for possible rational roots using the Rational Root Theorem, and then proceed to simplify or solve it.

Let me check for possible factors or solutions.The equation \( x^6 - 132x^5 + 7260x^4 - 212960x^3 + 3513840x^2 - 30921792x + 113379904 = 0 \) has a root at \( x = 22 \).

To further analyze the equation, you can factor it using this root or use numerical methods to find any additional roots.

Would you like to see the detailed factorization or explore more solutions?

Here are some related questions:
1. How can the Rational Root Theorem help in solving polynomial equations?
2. What are the next steps to fully factor the polynomial after finding a root?
3. What are the benefits of numerical methods in solving high-degree polynomials?
4. How does synthetic division work when factoring polynomials with known roots?
5. Can all polynomials of degree 6 be solved exactly by radicals?

**Tip:** Once a root is known, you can use polynomial division to factor the original equation and simplify further.

x^6 - 132x^5 +7260x^4 - 212960x^3 + 3513840x^2 - 30921792x + 113379904 = 0

Learn how to solve the degree-6 polynomial equation x^6 - 132x^5 + 7260x^4 - 212960x^3 + 3513840x^2 - 30921792x + 113379904 = 0 using the Rational Root Theorem, polynomial division, and numerical methods. This problem involves advanced algebraic concepts and is suitable for high school students in Grades 10-12 or college level.

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