Let the polynomial be such that never be zero.

 Consider .

Differentiate on each side with respect to .

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Differentiate on each side with respect to .

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

Therefore, need to prove that

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If .

Case 1:

If , then

Take the like terms onto one side.

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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.

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Since , then .

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

Condition is satisfied.

Therefore, Newtons method is converges.

True.