Approximate all real roots of the equation to two decimal places using Newton's Method.  X^4 = 125Must use Newtons Method.  Thanks!

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embizze eNotes educator| Certified Educator

Approximate the real roots of `x^4=125` to 2 decimal places:

Consider the graph of `x^4-125=0` :

It appears the roots are a littleabove 3 and below -3.

To use Newton's method,we find a"good" initial approximation, `x_1` . Then we determine a new approximation `x_(n+1)=x_n-(f(x_n))/(f'(x_n))` . We continue this process until `|x_n-x_(n+1)|` is within the desired accuracy; in this case to two decimal places.

(1) The root near 3: Let `x_1=3.5` . Note that `f'(x)=4x^3` . Then:




Since `|3.34374-3.34370|=0.00004` is within the desired accuracy we have thefirst root is approximately 3.34

(2) The root near -3. Let `x_1=-3.5`




So the second root is approximately -3.34