# Example of Newton's Method with Deflation and Horn

**Problem:** Find all solutions of

** Solution :** We’ll start with Newton’s method with p_{0} = 3. Set f(x) =
x^{4}+3x^{3}−12x−16 and evaluate both this function and its derivative via synthetic
division.

From the above table we are able to see that 2 is an
approximate solution , and

we also have the following factorization for our function :

We’ll simplify the problem by looking for zeros of the
last factor. This is called

deflation.

We now have another approximate solution and another factorization. Namely,

We’ll use the quadratic equation to find the last two zeros.

The two solution are −1.4929856 ± 1.3255958i

Errors accumulate in the coefficients during the deflation
process. We’ll use

Newton’s method with the original coefficients to refine the last three
solutions.

Again we use Horner’s rule for the calculations . The results are summarized

below.

Refinement of second solution : p_{2} = −2.00017, p_{3}
= −2.000000022.

Refinement of third solution :

The fourth solution must be the complex conjugate of the
third solution. Finally,

we list the four approximate solutions :

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