Apply the Implicit Function Theorem to find a root of polynomial 
Caculate the value of the real solution of the equation $x^7+0.99x-2.03$, and give a estimate for the error.

The hint is: use the Implicit Function Theorem. I dont know how to use the IFT in this case, I'm not familiarized with this.
I think in construct a function $F:\mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$ with some parameters of which one is the root. Maybe
$$F(c_1,c_2,c_3,x) = c_{1}x^7 + c_{2}x - c_{3}.$$
But I'm note sure about this. Can someone help me?

Let $F(x,y,z)=x^7+y\,x-z$; then $F(1,1,2)=0$. We have
$$
\frac{\partial F}{\partial x}=7\,x^6+y\implies\frac{\partial F}{\partial x}(1,1,2)\ne0.
$$
By the IFT, you can solve for $x$ in the equation $F(x,y,z)=0$ on a neighborhood of $(1,1,2)$. That is, there is a $C^1$ function $\phi(y,z)$ such that $\phi(1,2)=1$ and $F(\phi(y,z),y,z)=0$. What you want now is $\phi(0.99,2.03)$. You cannot obtain an exact formula, but you can find an approximation:
$$
\phi(0.99,2.03)\approx\phi(1,2)+\frac{\partial \phi}{\partial y}(1,2)(.99-1)+\frac{\partial \phi}{\partial z}(1,2)(2.03-2).
$$
You can find the values of the partial derivatives of $\phi$ from the IFT.
