I'm using this method:

First, write the polynomial in this form:

$$a_nx^n+a_{n-1}x^{n-1}+......a_2x^2+a_1x=c$$

Let the LHS of this expression be the function ##f(x)##. I'm gonna write the Taylor series of ##f^{-1}(x)## around ##x=0## and then put ##x=c## in it to get ##f^{-1}(c)## which will be the value of ##x##.

Since, ##f^{-1}(0)=0## here, so we've got the first term of our Taylor series as 0.

Now, the only thing that remains is calculating the derivatives of ##f^{-1}(x)## at ##x=0##.

I'm using the fact that $$\frac{d(f^{-1}(x)}{dx}=\frac{1}{f^{'}(f(^{-1}x))}$$

By differentiating this equation, we can get the second derivative of ##f^{-1}(x)## as:

$$\frac{d^2(f^{-1}(x))}{dx^2}=-\frac{1}{(f^{'}(f(^{-1}x)))^2}*f^{''}(f(^{-1}x))*f^{-1'}(x)$$

Similarly, we can get the other derivatives by further differentiation of this equation. Then we can evaluate all the derivatives at ##x=0## to get the Taylor series of ##f^{-1}(x)## and evaluate it at ##x=c## to get the value of ##x##.

1.Is this method correct?

2.Can something be done to make it better and remove the limitations?

First, write the polynomial in this form:

$$a_nx^n+a_{n-1}x^{n-1}+......a_2x^2+a_1x=c$$

Let the LHS of this expression be the function ##f(x)##. I'm gonna write the Taylor series of ##f^{-1}(x)## around ##x=0## and then put ##x=c## in it to get ##f^{-1}(c)## which will be the value of ##x##.

Since, ##f^{-1}(0)=0## here, so we've got the first term of our Taylor series as 0.

Now, the only thing that remains is calculating the derivatives of ##f^{-1}(x)## at ##x=0##.

I'm using the fact that $$\frac{d(f^{-1}(x)}{dx}=\frac{1}{f^{'}(f(^{-1}x))}$$

By differentiating this equation, we can get the second derivative of ##f^{-1}(x)## as:

$$\frac{d^2(f^{-1}(x))}{dx^2}=-\frac{1}{(f^{'}(f(^{-1}x)))^2}*f^{''}(f(^{-1}x))*f^{-1'}(x)$$

Similarly, we can get the other derivatives by further differentiation of this equation. Then we can evaluate all the derivatives at ##x=0## to get the Taylor series of ##f^{-1}(x)## and evaluate it at ##x=c## to get the value of ##x##.

1.Is this method correct?

2.Can something be done to make it better and remove the limitations?

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