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`
The passage contains a detailed mathematical problem involving the use of Newton's method to find the real roots of a polynomial equation. It includes the initial approximations, the iterative process, and the final answers to eight decimal places.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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- MULTIPLE_QUESTIONS
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system
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`
user
The passage contains a detailed mathematical problem involving the use of Newton's method to find the real roots of a polynomial equation. It includes the initial approximations, the iterative process, and the final answers to eight decimal places.

TYPE: SINGLE_QUESTION_WITH_ANSWER