An expression that remains unchanged when x and y are interchanged is called a symmetric expression. It can be expressed using x+y and xy, the elementary symmetric expressions.
Representation Using Elementary Symmetric Expressions
- x² + y² = (x + y)² − 2xy
- (x − y)² = (x + y)² − 4xy
- x³ + y³ = (x + y)³ − 3xy(x + y)
Example ①
When x + y = 5 and xy = 3, find (1) x² + y² and (2) x³ + y³.
Answer: (1) 25 − 6 = 19 (2) 125 − 45 = 80
When x + y = 5 and xy = 3, find (1) x² + y² and (2) x³ + y³.
Answer: (1) 25 − 6 = 19 (2) 125 − 45 = 80
Conjugate Irrational Numbers
The sum and product of x = 2 + √3 and y = 2 − √3 are rational numbers.
Example ②
When x = 2 + √3 and y = 2 − √3, find (1) x+y, (2) xy, and (3) x² + y².
Answer: (1) 4 (2) 4 − 3 = 1 (3) 16 − 2 = 14
When x = 2 + √3 and y = 2 − √3, find (1) x+y, (2) xy, and (3) x² + y².
Answer: (1) 4 (2) 4 − 3 = 1 (3) 16 − 2 = 14
Key Point
- x+y and xy are elementary symmetric expressions
- x² + y² = (x+y)² − 2xy is essential
- x³ + y³ = (x+y)³ − 3xy(x+y)
- For conjugate irrational numbers a ± √b, the sum is 2a and the product is a² − b
Practice Problems
- When x + y = 4 and xy = −2, find x² + y².
- When x = 3 + √2 and y = 3 − √2, find x+y and xy.
- When x + y = 6 and xy = 5, find x³ + y³.
Answers and Explanations
- Answer: 20
Explanation: 16 − 2×(−2) = 20. - Answer: x+y=6, xy=7
Explanation: The sum is 6, and the product is 9 − 2 = 7. - Answer: 126
Explanation: 216 − 90 = 126.