# 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...

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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