By replacing a common part with one letter, formulas can be used even for complicated expressions.
Replacing a Common Part
Let a + b = X and transform the expression into the form (X + c)(X − c) = X² − c².
Example ①
Expand (x + y + 1)(x + y − 1).
Answer:Let x + y = X. Then (X+1)(X−1) = X² − 1 = (x+y)² − 1 = x² + 2xy + y² − 1
Expand (x + y + 1)(x + y − 1).
Answer:Let x + y = X. Then (X+1)(X−1) = X² − 1 = (x+y)² − 1 = x² + 2xy + y² − 1
Changing the Order
Change the grouping order of (x+1)(x+2)(x−1)(x−2) to transform it into (x²−1)(x²−4).
Example ②
Expand (x+1)(x+2)(x−1)(x−2).
Answer:(x²−1)(x²−4). Let X = x². Then (X−1)(X−4) = X² − 5X + 4 = x⁴ − 5x² + 4
Expand (x+1)(x+2)(x−1)(x−2).
Answer:(x²−1)(x²−4). Let X = x². Then (X−1)(X−4) = X² − 5X + 4 = x⁴ − 5x² + 4
Key Points
- Simplify by replacing a common part
- Expansion is easier when you create the form (x+a)(x−a)
- Use a clever grouping order for four factors
Practice Problems
- Expand (a + b + 3)².
- Expand (x − y + 2)(x − y − 2).
- Expand (x + 2)(x + 4)(x − 2)(x − 4).
Answers and Explanations
- Answer:a² + 2ab + b² + 6a + 6b + 9
Explanation: Let a+b=X and expand (X+3)². - Answer:x² − 2xy + y² − 4
Explanation: Let x−y=X and use X² − 4 = (x−y)² − 4. - Answer:x⁴ − 20x² + 64
Explanation: (x²−4)(x²−16) = x⁴ − 20x² + 64.