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8181 is a root of f(x)=x2+6,561f(x) = x^2 + 6,561. Find the other roots of f(x)f(x).\newlineWrite your answer as a list of simplified values separated by commas, if there is more than one value.\newline______\newline

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Complex conjugate theoremright chevron icon

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Q. 8181 is a root of f(x)=x2+6,561f(x) = x^2 + 6,561. Find the other roots of f(x)f(x).\newlineWrite your answer as a list of simplified values separated by commas, if there is more than one value.\newline______\newline
  1. Identify Root: Since 8181 is a root, we can write one factor of f(x)f(x) as (x−81)(x - 81).
  2. Factor Polynomial: To find the other root, we need to factor the polynomial completely. We know that the constant term is 6,5616,561, which is 81281^2.
  3. Apply Difference of Squares: The polynomial can be written as f(x)=(x−81)(x+81)f(x) = (x - 81)(x + 81) because the difference of squares factorization is a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b).
  4. Set Equation to Zero: Setting f(x)f(x) to zero gives us the equation (x−81)(x+81)=0(x - 81)(x + 81) = 0.
  5. Solve for x: Solving for x, we get two solutions: x=81x = 81 and x=−81x = -81.

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