Let the polynomial be  such that  never be zero.

Consider .

Differentiate on each side with respect to .

.

Differentiate on each side with respect to .

Condition sufficient to produce convergence of Newtons method to a zero of  is that .

Therefore, need to prove that .

If .

Case 1:

If , then 

Take the like terms onto one side.

.

Since , then .

Left side in the above expression is square function which cannot be negative.

Case 2:

If , then .

Take the like terms onto one side and simplify the expression.

.

Since , then .

.

Left side in the above expression is square function which cannot be negative.

Condition is satisfied.

Therefore, Newtons method is converges.

True.