Given 2 points where a line and a curve cross, find the third point where line and the curve cross. The curve $y^2 = x^3 + 8$ contains the points $(1,-3)$ and $($$-7\over4$,$13\over8$$)$. The line through these two points intersects the curve in exactly one other point. Find this third point.
P.S. Is it possible to do this by polynominal division? I tried using $y = m(x-1)-3$ and equating line and curve and then dividing by $(x-1)$ and $(x + 7/4)$ but never got the correct answer.
 A: Since this is an elliptic curve, we have a nice way to compute the third point on the line (see the linked Wikipedia article for more information).
If the two known points are $P$ and $Q$ with the slope between them given as $s$ then the coordinates of the third point $R$ are given by
$$x_R=s^2-x_P-x_Q\hspace{20mm}y_R=y_P+s(x_R-x_P)$$
We have 
$$s=\frac{\frac{13}{8}+3}{-\frac{7}{4}-1}=-\frac{\frac{37}{8}}{\frac{11}{4}}=-\frac{37}{22}$$
Hence
$$x_R=\left(\frac{37}{22}\right)^2-1+\frac{7}{4}=\frac{433}{121}$$
and
$$y_R=-3-\frac{37}{22}\left(\frac{433}{121}-1\right)=-\frac{9765}{1331}$$
A: The better answer (using the properties of elliptic curves) was already posted by @Peter Woolfitt, but I would like to follow up on my comment, and on an addition to the question, namely:
"Is it possible to do this by polynominal division?" 
Answer, yes it is possible, the numbers are a bit messy so that would perhaps explain why you didn't get the correct answer. I cheated a bit again using a computer algebra system (reduce) to divide polynomials, but if one really insists this could be done by hand.  
So the line through the two given points has equation $y=\frac{-37x}{22}-\frac{29}{22}$. The curve could be written as $P(x,y)=x^3+8-y^2=0$. Plugging the equation of the line into the curve we get:  
$P(x,\frac{-37x}{22}-\frac{29}{22})=\dfrac{484x^3-1369x^2-2146x+3031}{484} = g(x)$. 
Then $\dfrac{g(x)}{x-1}= \dfrac{484x^2-885x-3031}{484}$.  
Also, $\dfrac{g(x)}{x+\frac74}= \dfrac{121 x^2- 554 x+ 433}{121}$. 
Finally, $\dfrac{g(x)}{(x-1)(x+\frac74)}=\dfrac{121 x- 433}{121}$.  
From the latter one would indeed correctly obtain that $x=\dfrac{433}{121}$. 
