Question : newton rhapson approximation

Hi

I have been doing the newton raphson approximation and the example i am reading at the moment says:

find the real root of the equation x^3 - 5x^2 - 3x - 4 = 0

The book says we need to find our starter value so we look for values where the sign of f(x) changes, which is thus near to where the function crosses the x axis and y = 0. Fair enough, i get that. But the books says

'apply the remainder theorem if the function is a polynomial' to find the starter value.

What's the remainder theorem got to do with it? Why don't you just plug in values of x into f(x) and see where the sign changes. Here f(0) = -4 and f(1) = -1 so there is a root between 0 and 1.

I'm not sure how that has got anything to do with the remainder theorem. If f(a) = 0 then that would tell me that x-a is a factor but I don't need that information hence i don't see what the remainer theorem has got to do with anything. Please can you tell me what I am 'not seeing'

thanks

Answer : newton rhapson approximation

>> 'apply the remainder theorem if the function is a polynomial' to find the starter value.
>> What's the remainder theorem got to do with it?

This simplest answer I've come up with is that the remainder theorem provides a way to calculate p( x ) for a number, x.

Do you have to use the remainder theorem to compute p( x )? No.

Since this was stated earlier, I think I'm at a loss as to what the question really is. If they never had the expression "remainder theorem", and just showed results for p(x) for x=0, x=±1, x=±2, etc.; then, would you have been OK with that? If so, then maybe there is no real question, since the remainder theorem is being used here to compute p(x).

And, if after using the algorithm and got an answer to 6 decimal places, you wanted double-check the result by taking your computed root, r, and then computing p(r) to see if it is extremely close to 0, then you could use the remainder theorem or just plug r into p(x) and find out what p(r) is.

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As an aside..
>> the question just asks to find the one root.
Actually, from your page, the question in Example 4 asks you to "find the real root". This suggests that there is only one real root with the other two roots being complex. In fact, there are three real roots.