Simon's Favorite Factoring Trick: A Clever Algebraic Technique

Simon's Favorite Factoring Trick (SFFT) is a useful algebraic manipulation method to factor certain expressions by adding and subtracting strategically. This trick is particularly useful in problem-solving competitions like the AMC (American Mathematics Competitions) and is named after Simon Rubinstein-Salzedo, who popularized it on the Art of Problem Solving (AoPS) forums.

1. Understanding Simon’s Favorite Factoring Trick

The trick is based on the observation that you can transform an expression of the form:


into a factored form:


The key step is to identify missing terms and add/subtract them to complete the factorization.

2. The General Form of SFFT

Given an expression of the form:


you aim to rewrite it as:


by cleverly adding and subtracting the correct constant. Specifically:


where you choose and such that:


3. Example 1: Solving a Classic SFFT Problem

Solve for the integer solutions of the equation:

• Step 1: Rearrange the equation:
  
  
• Step 2: Identify the "missing term" for factoring.
  
  We want to rewrite this expression to match the form:
  
  
  So, we add and subtract 20:
  
  
  
• Step 3: Find all factor pairs of 36:
  
  
• Step 4: Solve for and :
  
  
  For example:
  
  
  Thus, one solution is .
  

4. Example 2: Factoring a Quadratic Expression

Factor the expression:

• Step 1: Rearrange the terms:
  
  

5. Python Code: Implementing SFFT for Factoring

Here's a simple Python program to apply Simon’s Favorite Factoring Trick:

def sfft(x, y, a, b, c):
    print(f"Original expression: {x}y + {a}x + {b}y + {c}")
    missing_term = a * b
    print(f"Adding and subtracting {missing_term}")
    print(f"Factored form: ({x} + {b})({y} + {a}) - {missing_term} + {c}")

sfft('x', 'y', 4, 5, 20)

6. Why Does Simon’s Favorite Factoring Trick Work?

The trick relies on "completing the product." We transform a linear combination into a product by adding and subtracting a strategically chosen term:


By controlling and , you can rewrite the expression in factored form.

7. Applications of Simon’s Favorite Factoring Trick

• Competition Math: Appears frequently in the AMC, AIME, and other math competitions.
• Number Theory: Useful for solving Diophantine equations and integer solutions.
• Algebraic Manipulation: Helps transform complex expressions into simpler factored forms.

8. Variations and Extensions of SFFT

• Higher Degree Polynomials: Can be extended to factor cubic and quartic expressions with similar techniques.
• Negative Coefficients: Works even when coefficients are negative by adjusting the sign of the added term.
• Geometric Interpretation: Visualize SFFT as rearranging areas of rectangles to form perfect squares.

9. Fun Facts About Simon’s Favorite Factoring Trick

• Simon Rubinstein-Salzedo introduced the trick on the AoPS forums.
• It simplifies many otherwise tricky algebraic problems.
• The trick generalizes ancient methods used by mathematicians like Diophantus.

10. Conclusion

Simon’s Favorite Factoring Trick is a powerful and elegant technique for transforming and factoring algebraic expressions. By adding and subtracting the right terms, you can unlock new approaches to solving equations efficiently.

References

• AoPS: Simon’s Favorite Factoring Trick
• Wikipedia: Factoring in Algebra
• Zeitz, P. The Art and Craft of Problem Solving (2007).
• Larson, E. Problem-Solving Through Problems (1983).
• Kline, M. Mathematical Thought from Ancient to Modern Times (1972).