Quiz 2 Solution

On our quiz, we had the following problem:

Generate a cubic polynomial of the form f(z)=z^3 + p z + (-1)^s q by choosing

• p to be the position in the alphabet of the first letter in your first name,
• q to be the position in the alphabet of the last letter in your last name, and
• s to be the position in the alphabet of your middle initial.

For example, my name is Mark C.~mcclurE; thus, my cubic is f(z)=z^3 + 13 z - 5.

1. Explain why this process always generates a polynomial with precisely one root in the upper half plane.
2. Apply Newton’s method to find a good decimal approximation to the root. Be sure to
   a) Find the Newton’s method iteration function and type it out nicely,
   b) use a picture to find a good starting point from which to iterate, and
   c) perform the iteration to find the root precise to 10 decimal places.

Solutions might follow.

Solution to part 1.

We know that every polynomial of degree n has exactly n roots, counted according to multiplicity.

Thus, every cubic has three roots, counted according to multiplicity. Furthermore, our cubic has at least one real root, since it is continuous with

By construction, f'(x) = 3x^2+p>0 for all real x. Thus, the function is always increasing so it can have only one root, which must be simple. The two other roots are complex and must be complex conjugates. That gives us exactly one root in the complex plane.

Solution to part 2.

Since my function is f(z)=z^3+13z-5, my Newton’s method iteration function is:

I can iterate that from z_0=10i using this Sage code:

f(x) = x^3 + 13*x - 5
N(x) = x - f(x)/diff(f(x),x)
xi = 10j  # start at x=10i
for i in range(10):
    xi = N(xi)
    print(xi)

This prints a sequence that converges to