Question

Use the Newton-Raphson iteration procedure to find the root of 𝑥^3−6𝑥^2+9𝑥+1 = 0

Answer 1

Hi

Answer 2

You will need at least one approximation to the root(s) to get you started.

Differentiating we get 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x-3)(x-1) = 0
so x = 3 and x = 1 are our relative maxima and minima.
P(3) = 1 and P(1) = 5

So we anticipate it will look something like this:


So expect the root to occur for x < 0.
But sub in points and see what x gets the cubic closer to 0.
If your x becomes more negative but the cubic becomes more negative, we need to head in the opposite direction.

Once you find a suitable 'alpha', then apply Newton's law to it.