Question

两个不同的四次多项式函数 $y = p(x)$ 和 $y = q(x)$ 的图像最多有多少个交点，若它们的首项系数都为 $1$ ？

What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $y = p(x)$ and $y = q(x),$ each with leading coefficient $1?$

$1$

$2$

$3$

$4$

$8$

答案：C

知识点：多项式交点计数

难度评级：1510

解答：

交点的 $x$ 坐标是 $p(x) - q(x)$ 的根。由于两个四次多项式首项系数都为 $1$ ，相减时 $x^4$ 项抵消，所以 $p(x) - q(x)$ 次数至多为 $3$ ，最多有 $3$ 个实根。三个交点可以实现。 $p(x) = x^4$ $q(x) = x^4 - x(x-1)(x+1)$ ， $x = -1, 0, 1$ 。

所以正确答案是 C。

The $x$ -coordinates of the intersection points are the roots of $p(x) - q(x).$ Because both leading coefficients are $1,$ the $x^4$ terms cancel, so $p(x) - q(x)$ has degree at most $3$ and therefore at most $3$ roots. The bound is attainable, for example by taking $p(x) = x^4$ and $q(x) = x^4 - x(x-1)(x+1),$ whose graphs intersect at $x = -1, 0, 1.$

Thus, the correct answer is C.

1999 AMC 12 第 12 题

答案：C

难度评级：1510

解答：

其他年份的第 12 题