(1)^4 - 4a(1)^3 + (3a + b)(1)^2 + (4a - 7b + c)(1) + 60 = 0

Let's simplify and solve the given polynomial equation:

\[ (1)^4 - 4a(1)^3 + (3a + b)(1)^2 + (4a - 7b + c)(1) + 60 = 0 \]

1. Simplify each term involving 1:
   - \((1)^4 = 1\)
   - \(4a(1)^3 = 4a\)
   - \((3a + b)(1)^2 = 3a + b\)
   - \((4a - 7b + c)(1) = 4a - 7b + c\)

2. Substitute these into the equation:

\[ 1 - 4a + 3a + b + 4a - 7b + c + 60 = 0 \]

3. Combine like terms:

\[ 1 + (3a - 4a + 4a) + (b - 7b) + c + 60 = 0 \]
\[ 1 + 3a - 4a + 4a + b - 7b + c + 60 = 0 \]
\[ 1 + 3a + b - 7b + c + 60 = 0 \]
\[ 1 + 3a - 6b + c + 60 = 0 \]

4. Simplify further:

\[ 3a - 6b + c + 61 = 0 \]

5. Solve for \(c\):

\[ c = -3a + 6b - 61 \]

This is the relationship among \(a\), \(b\), and \(c\) derived from the given polynomial equation.

Would you like more details or have any questions about the solution?

Here are 8 relative questions expanding on the provided information:

1. What is a polynomial equation, and how is it generally structured?
2. How do you identify and combine like terms in polynomial equations?
3. What are the common methods for solving polynomial equations?
4. How does the degree of a polynomial affect its graph and roots?
5. Can you explain the significance of coefficients in polynomial equations?
6. How do you factorize polynomials, and what are some useful techniques?
7. What are the real-life applications of solving polynomial equations?
8. How do you determine the roots of a polynomial equation using the Rational Root Theorem?

**Tip:** Always double-check your work for arithmetic errors when simplifying equations to ensure accuracy in your solutions.

Learn how to solve the polynomial equation (1)^4 - 4a(1)^3 + (3a + b)(1)^2 + (4a - 7b + c)(1) + 60 = 0 with a detailed step-by-step solution. This problem involves simplifying algebraic expressions and finding relationships among variables a, b, and c. Suitable for advanced high school students and beyond.

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