Need Help finding the Multiplicity Root of equation?

Question

Hi all,
I’m having problems with the following question.
“Find the multiplicity root of the equation x^3 – x^2 – x +1 = 0 near x0 = 0.9 the answer of the problem that I have been given is 1.773 ~ 2.0

I can’t get this answer or anything near this. Could anyone please tell me what I am doing wrong
I’m using the formula
[f ‘ (x0)] ^2
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[f ‘ (x0)] ^2 – f ‘’ (xo) f’(xo)

To get the value m. Please tell me if this is the right way to proceed with this question. Id appreciate show me the steps being used here.

Thanks in advance

Answer 1

You're making this way too complicated. Why don't you just factor by grouping and find the roots that way:

x³ – x² – x +1 = 0
x² (x-1) -1(x-1) = 0
(x² -1)(x-1) = 0
(x-1)(x+1)(x-1) = 0
(x-1)² (x+1) = 0

Therefore x = 1 has multiplicity two and x = -1 has multiplicity one.

Answer 2

I would first search for a root and simplify using synthetic division: for example, x=1 is a root--for factor out (x-1). Doing this give you x^2-1: this in tern also has x=1 as a root. Therefore, this equation has root x=1 with multiplicity 2.