Simon's Favorite Factoring Trick
The General Statement
The general statement of SFFT is: . Two special common cases are: and .
The act of adding to in order to be able to factor it could be called "completing the rectangle" in analogy to the more familiar "completing the square."
This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually and are variables and are known constants. Also, it is typically necessary to add the term to both sides to perform the factorization.
Fun Practice Problems
- Two different prime numbers between and are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
- are integers such that . Find .
- The integer is positive. There are exactly 2005 ordered pairs of positive integers satisfying:
Prove that is a perfect square. (British Mathematical Olympiad Round 2, 2005)
Enjoy the problems!