There is also a formula for the square of a sum with three or more terms.
Formula for the Square of Three Terms
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
How to remember it: “the sum of the squares of each term + twice the sum of the products of every distinct pair of terms”
Example ①
Expand (x + 2y + 3)².
Answer:x² + 4y² + 9 + 4xy + 12y + 6x
Expand (x + 2y + 3)².
Answer:x² + 4y² + 9 + 4xy + 12y + 6x
Example ②
Expand (a − b + c)².
Answer:a² + b² + c² − 2ab − 2bc + 2ca
Expand (a − b + c)².
Answer:a² + b² + c² − 2ab − 2bc + 2ca
Key Points
- (a+b+c)² = a²+b²+c² + 2(ab+bc+ca)
- When terms have signs, include those signs when multiplying each pair of terms
Practice Problems
- Expand (x + y + 4)².
- Expand (2a − b + 3)².
- Expand (x − 2y − 5)².
Answers and Explanations
- Answer:x² + y² + 16 + 2xy + 8y + 8x
Explanation: Substitute into the formula. - Answer:4a² + b² + 9 − 4ab − 6b + 12a
Explanation: Apply the formula to 2a, −b, and 3. - Answer:x² + 4y² + 25 − 4xy + 20y − 10x
Explanation: Apply the formula to x, −2y, and −5.