Analysis of Newton's Method

Question

I want to use Newton's method to find the roots of $$ 2x^3 − 9x^2 + 12x + 15$$ for $x=3$, $x<-3$, and $x>3$. I see that if $x=3$, the Newton iterates do not converge. How do I handle the cases when $x>3$ and $x<3$. I don't see an obvious behavior in the iterates after going through a few steps with guesses in $x<3$ or $x>3$. Thanks.

Answer 1

Newton's method is only guaranteed to converge to a solution (that satisfies $f'(x) \neq 0$) if you start 'close enough'.

In this case the polynomial $p$ has exactly one real root, and since $p(-1) = -18, p(0) = 5$, we see that it lies in $(-1,0)$.

Try starting at $-{1 \over 2}$ for example.