Question

美国大学入学数学测试...

1.Which of the following polynomials has a root between 0 and 1 ?
A.x^3-4x^2-3x-1
B.x^3+4x^2-3x-1
C.x^3-4x^2+3x+1
D.x^3+4x^2-3x+1
E.None of the above
2.If A and B are real numbers and x^3-ax+b has a root equal to 2-3i,then a second root is
A.-2-3i
B.2+3i
C.-2+3i
D.3
E.None of the above

Answer 1

1.Answer: B.x^3+4x^2-3x-1
Ideas: Let f(x) = x^3+4x^2-3x-1. Since f(0) = -1 < 0, and f(1) = 1> 0, by continuity the polynomial has a root between 0 and 1.

2. Answer: B.2+3i
Ideas: If a-bi is a root, then its conjugate a+bi must be also a root.

美国高中数学教师

Answer 2

1. 用画图计算器画图，与X轴交点在0与1之间的就是选项。。

抱歉啊。我这里没有画图计算器，但是这是我能想到的最方便快捷的解法。

2. 选B，因为与题中提到的解互为conjugate root。就是实数前符号不变，虚数前变正负。