A polynomial function P has zeros -3,3/2,and 8. Which of the following polynomial functions could define P?

多项式函数P存在 -3,3/2,8三个零点，则P的多项式函数是以下哪一个?

A. P(x)=-3(x-3/2)(x-8) B.P(x)=-(x-3)(x+3/2)(x+8) C. P(x)=(x+3)(3x-2)(x-8) D. P(x)=(x+3)(2x-3)(x-8)

答案：D

解析：Recall that if K is a zero of a polynomial function defined as y=f(x), then x-k is a factor of f.

要牢记，如果K是函数y=f(x)的零点，则x-k是函数f的一个因式

Since the polynomial function P has the zeros -3,3/2,and 8,it follows that (x-(-3)),(x-3/2),and (x-8) must be factors of P.

既然该多项式函数P有-3,3/2,8三个零点，则可得到(x-(-3)),(x-3/2), (x-8)都是P的因式。

Therefore, we can define P as P(x)=a(x+3)(x-3/2)(x-8), where a is a nonzero constant.

所以，我们可以定义函数P为P(x)=a(x+3)(x-3/2)(x-8), 其中a是非零常量。

A constant factor, such as a, does not affect the zeros of the polynomial function. In order to rewrite the equation with integral coeffecients, let a=2.

最后一步，用整数系数改写一下该方程。

If a=2, it follows that

P(x)=a(x+3)(x-3/2)(x-8)

=2(x+3)(x-3/2)(x-8)

=(x+3)(2x-3)(x-8).

so the polynomial that could define P is P(x)=(x+3)(2x-3)(x-8).

得到最终的可能结果之一为P(x)=(x+3)(2x-3)(x-8).