Using Newton-Raphs
f (x) = 2x3 + x2 - x + 1 accurate to within 10-4

9514 1404 393

  -1.2338

Step-by-step explanation:

The graph shows the one real root is about -1.234. The Newton's method iterator is ...

  new x = x - f(x)/f'(x) . . . . . where f'(x) is the derivative of f(x)

Many graphing calculators provide a derivative function, so the iteration equation can be evaluated more or less directly by the calculator. In the attached, we have shown a table with three iterations of the root. After the second iteration, the root is good to 4 decimal places.

  x ≈ -1.2338

the mistake in the pattern is that :

b. the decimal point was moved one too many place values each time

the correct pattern could be made by :

b. moving the decimal point to the right the same numbers of place values as the exponents