奧數問題(explain plz)?

chung

2016-01-22 2:52 pm

1.if the equation x^3+ax^2-17x+1=0(where a is an integer) has rational solution,find the greatest possible value of a. Ans:15

2.it is known that the equation x^3-9x^2+15x+c=0(where c is a constant) has a multiple root.Find the sum of all possible values of that multiple root. Ans:6

回答 (1)

用戶2053

2016-01-22 5:22 pm

✔ 最佳答案

1)
x³ + ax² - 17x + 1 = 0
令 rational x = p/q 且 p與q互質, 明顯 x ≠ 0 故 p、q 為非0整數.
p³/q³ + ap²/q² - 17p/q + 1 = 0
p³ + ap²q - 17pq² + q³ = 0
a = (17pq² - p³ - q³) / (p²q) 為整數, 且p、q為整數,
故(17pq² - p³ - q³) / p = 17q² - p² - q³/p 及 (17pq² - p³ - q³) / q = 17pq - q² - p³/q 都是整數.
從而 q³/p 及 p³/q 皆為整數, 但p與q互質故而 p = q = 1 , 即有理根 x = 1 , 代回原方程得 1 + a - 17 + 1 = 0,
a = 15.

2)
Let a be the multiple root while b is another root.
2a + b = 9 and a² + 2ab = 15
a² + 2a(9 - 2a) = 15
a² - 6a - 5 = 0
Sum of 2 possible values of that multiple root a = 6.

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