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
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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`
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