Even complicated expressions can be reduced to basic formulas by substitution.
Replacing a Common Part
Example ①
Factor (x + y)² − 5(x + y) + 6.
Answer:Let x + y = X. Then X² − 5X + 6 = (X − 2)(X − 3).
Substitute back for X: (x + y − 2)(x + y − 3)
Factor (x + y)² − 5(x + y) + 6.
Answer:Let x + y = X. Then X² − 5X + 6 = (X − 2)(X − 3).
Substitute back for X: (x + y − 2)(x + y − 3)
Biquadratic Expressions (x⁴ Form)
For x⁴ + ax² + b, let x² = X and factor it as a quadratic expression.
Example ②
Factor x⁴ − 13x² + 36.
Answer:X = x²: X² − 13X + 36 = (X − 4)(X − 9) = (x² − 4)(x² − 9) = (x + 2)(x − 2)(x + 3)(x − 3)
Factor x⁴ − 13x² + 36.
Answer:X = x²: X² − 13X + 36 = (X − 4)(X − 9) = (x² − 4)(x² − 9) = (x + 2)(x − 2)(x + 3)(x − 3)
Key Points
- Replace a common part with one letter
- For a biquadratic expression, let x² = X
- Continue factoring until the expression cannot be factored any further
Practice Problems
- Factor (x − 1)² − 4(x − 1) + 3.
- Factor x⁴ − 5x² + 4.
- Factor x⁴ − 10x² + 9.
Answers and Explanations
- Answer:(x − 2)(x − 4)
Explanation: Let X = x−1. Then X²−4X+3 = (X−1)(X−3) = (x−2)(x−4). - Answer:(x+1)(x−1)(x+2)(x−2)
Explanation: (x²−1)(x²−4) → then use the difference of squares again. - Answer:(x+1)(x−1)(x+3)(x−3)
Explanation: (x²−1)(x²−9).