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63. Suppose that L is the tangent line at to the graph of the cubic equation . Find the x-coordinate of the point where L intersects the graph a second time.

First, we find the derivative of y:

This gives us the slope of L at . Next we designate the point as the point where L is tangent to y(Note: is treated as a constant from here on out).

Now, using the point-slope form of a line, we define L:

We can write in terms of using the original equation:

Then,

Now that we have the above formula for the tangent line L, we set it equal to the original cubic equation and find all the solutions:

To factor the above we will use synthetic division. We already know that is a factor, because is where L is tangent to the above.








Since the remainder is 0, this confirms that is a factor.

Thus, L crosses at


Given:

Prove:



Given:

Where c is a constant, prove:



Given:

Prove: