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New points from old

Published online by Cambridge University Press:  23 January 2015

Stan Dolan
Stan Dolan*
Affiliation:
126A Harpenden Road. St. Albans AL3 6BZ
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Extract

Given two rational points on the graph of a cubic polynomial with rational coefficients, it is well known that one can construct another rational point by finding the third intersection point of the curve with the chord through the two points. For example, consider the two points A (2, 15) and B(0, 3) on y = 2x3 − x2 + 3.

The chord through A and B has equation y = 6x + 3 and this intersects the curve again at the point P, where

This cubic equation has three roots corresponding to A, B and the new point P,

P is the point

Type
Articles
Information
The Mathematical Gazette , Volume 96 , Issue 537 , November 2012 , pp. 492 - 498
Copyright
Copyright © The Mathematical Association 2012

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