2道英文的数学题,1)If f (x) = ax^2 + bx + c; f (2012) = 0; f (2010) = 0 and f (0) = 8,088,240then for what value of x is f a maximum?What is the maximum value of f 2)What is the remainder when2011x^2010+ 2009x^1005+ 2005x^201+ 2003x^67+ 2001 is divided by x + (Don’t use synthetic division!)

问题描述：

2道英文的数学题,
1)If f (x) = ax^2 + bx + c; f (2012) = 0; f (2010) = 0 and f (0) = 8,088,240
then for what value of x is f a maximum?What is the maximum value of f
2)What is the remainder when2011x^2010+ 2009x^1005+ 2005x^201+ 2003x^67
+ 2001 is divided by x + (Don’t use synthetic division!)

答

翻译：1 设f (x) = ax^2 + bx + c，且有 f (2012) = 0; f (2010) = 0 and f (0) = 8,088,240
求 x取何值时函数f有最大值，并求出最大值的f
2 2011x^2010+ 2009x^1005+ 2005x^201+ 2003x^67+ 2001除以 x+1的余式（不要使用多项式除法）
解答，第一题应该是有问题啊，没有最大值可求啊，确定符号什么都没打错么？
第二题让我想想。

答

1\ x=2011 f(x)取到最值
由韦达定理,-b/a=4022 c/a=4044120
c=f(0)=8088240
所以a=2 b=-8044
最小值fmin=f(2011)=2(2011-2012)(2011-2010)=-2
2\ 将x=-1带入,原式=2011-2009-2005-2003+2001=-2005
所以原式加上2005后,就含有因式x+1
又因为原式除以x+1的结果必然是一个常数,所以这个常数是-2005,这就是余数

2道英文的数学题,1)If f (x) = ax^2 + bx + c; f (2012) = 0; f (2010) = 0 and f (0) = 8,088,240then for what value of x is f a maximum?What is the maximum value of f 2)What is the remainder when2011x^2010+ 2009x^1005+ 2005x^201+ 2003x^67+ 2001 is divided by x + (Don’t use synthetic division!)

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