Rationnal roots with constant=0 ???

My bad I just noticed I didn't need Gauss on this one.

But the constant = 0.

How can you solve this if you can't even find the potential rationnal roots, as 0 cannot be divided?

If I understand what you are asking (I'm not entirely sure I do!) then when you divide by x to find the other roots, you are dividing by a non-zero value - because you are looking for values of x that are not zero.  The same is true when you divide by any of the other potential roots - although you seem to be dividing by zero (because x-x_a, say, gives a root at x_a) you are not, because now you are looking for values of x that are anything but x_a.

Not sure if that addresses your question!

Honestly I didn't fully get what you meant there. But my question is this:

If you say x=0 as in "x" is a potential rationnal root of the polynomial, wouldn't it mean that you have divided said polynomial by 0 in the first place?

But obviously we can't divide by 0...

So yah, the x=0, I just don't get it.

I am so sorry.

I just noticed I forgot a "+1" at the end of the polynomial.

That would be :

x^3 + 3x^2 + 2x + 1

The polynomial  x^3 + 3x^2 + 2x + 1 has just one real root, and doesn't factorise nicely - see the graph below:

Thanks a lot!

Although, I have the find some exact coords:

I found x=0 which is (0,1)

But I have to find y=0

That is why I'm trying to find it's root.

I also have to find the minima and maxima.

You will need a numerical method to find the root. One possibility is the Newton-Raphson method:

y(x) = x^3 + 3x^2 + 2x + 1

dy/dx(x) = 3x^2 + 6x + 2

Start with an initial guess of x₀ (say, -2.5)

Then use the following iterative process:  x_n+1 = x_n - y(x_n)/dy/dx(x_n) repeatedly until you get a sufficiently accurate result.

To find the turning points (not the maximum, that's clearly infinity, nor the minimum, that's clearly -infinity) set dy/dx equal to zero and find its roots:

3x^2 + 6x + 2 = 0

This will give you the values of x where the turning points occur.

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I'm sorry I don't understand the formula you're using there.

The exact task is to approximate the local max and local min (I didn't mean the - or + infinite parts) and to approximate the coords of intersection points.

Like at this point in time I'm not "supposed" to know how to find the exact coords, as it's part a math class. 

Only tools I have is finding roots the basic way if a>1 or Gauss (wich doesn't work because the constant here is 1).

So as of now I have this:

(0,1)

approx: (-2.25,0)

Local (or relative) minimum: approx (0.4, 0.55)

Local (or relative) maximum: approx (-1.5 , 1.375)

The Newton Raphson formula is an iterative one. You plug your initial guess x₀ into the right hand side to get a better estimate x₁.  You then plug x₁ into the right hand side to get an even better estimate x₂.  You repeat this process a few times until two successive estimates are the same to within the accuracy you are working to.

To get the turning points use the quadratic equation to get the roots:

x_min = (-6 - sqrt(6^2 - 4*3*2))/(2*3)

x_max = (-6 + sqrt(6^2 - 4*3*2))/(2*3)

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An alternative method of getting an approximate root of your cubic is to note that, from the graph it is clear that the solution lies between -2 and -2.5. Keep guessing values in between and try them in the cubic to see how close to zero you can get. The human brain is good at this sort of guesswork, and it probably won't take many guesses to get several decimal places of accuracy!

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