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

Asked on by gulusoy

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mathsworkmusic | (Level 2) Educator

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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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