Calculus: The tangent line intersects a curve at two points. Find the other point. The line tangent to $y = -x^3 + 2x + 1$ when $x = 1$ intersects the curve in another point.  Find the coordinates of the other point.
This was never taught in class, and I have a test on this tomorrow.  This question came off of my test review worksheet, and I don't understand how to solve it.  The answers are on the back, and for this one it says the answer is (-2,5), but I don't understand how to get that.
I did the derivative and substituted 1 for x to get the slope of the line:
$y = -x^3 + 2x + 1$
$y' = -3x^2 + 2$
$y' = -3 + 2$
$y' = -1$
I don't know where to go from here.
We first describe the normal computational approach, and then a more conceptual approach. 

Computational approach: You can find the equation of the tangent line. After some work you will end up with $y=3-x$.

Substitute $3-x$ for $y$ in the equation of the curve. We end up with a cubic equation $x^3-3x+2=0$. 

This equation has $x=1$ as a root.  Divide $x^3-3x+2$ by $x-1$. You will get $x^2+x-2$, which factors as $(x-1)(x+2)$. That gives the $x=-2$.

Conceptual approach: Just  imagine finding the equation of the tangent line, as we did above, but don't do the actual work. When we substitute for $y$ in the equation of the curve,  we  end up with a cubic equation of the shape $P(x)=x^3+ax+b=0$. 

Since we are not doing the work, we won't find $a$ or $b$. 

The tangent line at $x=1$ kisses the curve at $x=1$.  Thus the equation $P(x)=0$ has $x=1$ as a double root. So the sum of the roots is $1+1+w$, where $w$ is our mystery number.

The sum of the roots of $P(x)=0$ is the negative of the coefficient of $x^2$, in this case, $0$. So $1+1+w=0$ and therefore $w=-2$.
x=1

y=-x3+2x+1

y=2

m=dy/dx=-3x2+2

x=1

m=-1

y-2=-1(x-1)

y=3-x

3-x=-x3+2x+1

0=-x3+3x-2

let x=1

0=0

:: x-1 is a factor

(x-1)(-x2-x+2)=-x3+3x-2

-x2-x+2=0

x=1 , x=-2

0=(x-1)(x-1)(x+2)

x=1(N.A) , x=-2 , x=1(N.A)

y=5

PLEASE, TRY TO UNDERSTAND