What is the sum of all the roots of the equation?
Consider the equation (1-x)^{4}+(5-x)^{4}=82.
- A. 24
- B. 12 ✓
- C. 10
- D. 6
Correct Answer: B. 12
Explanation
Expanding the equation (x-1)^4 + (x-5)^4 = 82 gives 2x^4 - 24x^3 + \dots = 0. By Vieta's formulas, the sum of all roots (both real and complex) is given by the coefficient of x^3 divided by the coefficient of x^4 multiplied by -1, i.e., -(\frac{-24}{2}) = 12.
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