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Is it possible to obtain all the real roots of a polynomial equation using Newton's method? If it is, then is it done by giving different approximations close to each one of the real roots?

For example: If the roots of the polynomial equation are $1$ and $-2$, should we replace $x$ with both $\frac{4}{3}$ and $-\frac{9}{4}$ to obtain 2 roots, or does it only give one of them?

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Newton's method is local, and returns only one root at a time. You'll have to know in advance how many roots there are and where approximately they are located in order to get them all by Newton's (and Raphson's) method.

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