標題:
F.4 AM
發問:
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1a)Write down the expansions of (a+b)^3 and (a+b)^5. b)If a +b=p and ab=q,express a^3+b^3 and a^5 and b+5 in terms of p and q. c)If a+b=-3 and a^5+b^5=-123,obtain the possible values of ab and derive the possible values of a^3+b^3. 2)If ax^2+bx+c=0 has a double root and b0,show that the root is -2c/b
最佳解答:
1a) (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 (a+b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5 1b) a^3 + b^3 = (a+b)^3 - 3a^2b - 3ab^2 = p^3 - 3aq - 3bq = p^3 - 3q(a+b) = p^3 - 3pq = p(p^2 - 3q) a^5 + b^5 = (a+b)^5 - 5a^4b - 10a^3b^2 - 10a^2b^3 - 5ab^4 = p^5 - 5a^3q - 10aq^2 - 10bq^2 - 5b^3q = p^5 - 5q(a^3 + 2aq + 2bq + b^3) = p^5 - 5q(a^3 + b^3 +2pq) = p^5 - 5q(p^3 - pq) = p^5 - 5p^3q + 5pq^2 1c) (a^5+b^5) = -123 , a^5 + b^5 = p^5 - 5p^3q + 5pq^2 -123 = -233 - 5(-3)^3q + 5(-3)q^2 120 = 135 q -15q^2 0 = 15q^2 - 135q + 120 q = 8 OR 1 ab = 8 OR 1 when ab = 8 , a^3 + b^3 = p(p^2 - 3q) =-3(9-24) = 45 when ab = 1 , a^3 + b^3 = p(p^2 - 3q) = -3(9 - 3) = -18 2007-11-04 21:02:49 補充： 2) ax^2 bx c=0 delta = b^4 - 4ac BECAUSE IT HAS DOUBLE ROOT b^2 - 4ac = 0 b^2 = 4ac b^2/4c = a A B = -b/a AB = c/aBECAUSE IT HAS DOUBLE ROOTA B = 2A = -b/aAB = A^2 = c/a2A = -b/a = -b/b^2/4c = -4c / b A = -2c / b the root is -2c/b
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