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`