text: Use Newton's method to find all the real roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. x6 − x5 − 5x4 − x2 + x + 6 = 0
  
  First draw the graph using a calculator or computer
  
  Approximate the four roots (where the graph cuts the x axis) giving something like
  
  `x= -1.8,-1.1,1.1,2.8`
  
  Newton's method involves refining our estimates using the recursive formula
  
  `x_(n+1) = x_n - f(x_n)/(f'(x_n))`
  
  Now, we have `f(x) = x^6-x^5 -5x^4 -x^2+x+6`
  
  `implies` `f'(x) = 6x^5 -5x^4 -20x^3 -2x +1`
  
  For each initial estimate we get a sequence of improving approximations:
  
  1) ` `x0 -1.8
  
  x1 -1.76907576
  
  x2 -1.76617802
  
  x3 -1.76615375
  
  x4 -1.76615375
  
  2) x0 -1.1
  
  x1 -1.08064658
  
  x2 -1.08057763
  
  x3 -1.08057763
  
  3) x0 1.1
  
  x1 1.05017360
  
  x2 1.04708065
  
  x3 1.04706901
  
  x4 1.04706901
  
  4) x0 2.8
  
  x1 2.78786233
  
  x2 2.78761037
  
  x3 2.78761026
  
  x4 2.78761026
  
  Therefore we have real 4 solutions (the other two are complex)
  
  `x = -1.76615375``-1.08057763``1.04706901`  and `2.78761026`
explanation: The document provides a detailed explanation of using Newton's method to find the real roots of a polynomial equation. It includes the initial step of graphing the function to estimate the roots, followed by the application of Newton's method to refine these estimates to eight decimal places. The document also lists the final approximations of the roots.
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distilabel_metadata.content: Use Newton's method to find all the real roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. x6 − x5 − 5x4 − x2 + x + 6 = 0
  
  First draw the graph using a calculator or computer
  
  Approximate the four roots (where the graph cuts the x axis) giving something like
  
  `x= -1.8,-1.1,1.1,2.8`
  
  Newton's method involves refining our estimates using the recursive formula
  
  `x_(n+1) = x_n - f(x_n)/(f'(x_n))`
  
  Now, we have `f(x) = x^6-x^5 -5x^4 -x^2+x+6`
  
  `implies` `f'(x) = 6x^5 -5x^4 -20x^3 -2x +1`
  
  For each initial estimate we get a sequence of improving approximations:
  
  1) ` `x0 -1.8
  
  x1 -1.76907576
  
  x2 -1.76617802
  
  x3 -1.76615375
  
  x4 -1.76615375
  
  2) x0 -1.1
  
  x1 -1.08064658
  
  x2 -1.08057763
  
  x3 -1.08057763
  
  3) x0 1.1
  
  x1 1.05017360
  
  x2 1.04708065
  
  x3 1.04706901
  
  x4 1.04706901
  
  4) x0 2.8
  
  x1 2.78786233
  
  x2 2.78761037
  
  x3 2.78761026
  
  x4 2.78761026
  
  Therefore we have real 4 solutions (the other two are complex)
  
  `x = -1.76615375``-1.08057763``1.04706901`  and `2.78761026`
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document provides a detailed explanation of using Newton's method to find the real roots of a polynomial equation. It includes the initial step of graphing the function to estimate the roots, followed by the application of Newton's method to refine these estimates to eight decimal places. The document also lists the final approximations of the roots.
  
  Label: question-and-answer