AMC12 2000 A

AMC12 2000 A · Q22

AMC12 2000 A · Q22. It mainly tests Polynomials, Algebra misc.

The graph below shows a portion of the curve defined by the quartic polynomial $P(x) = x^4 + ax^3 + bx^2 + cx + d$. Which of the following is the smallest?

下图显示了由四次多项式 $P(x) = x^4 + ax^3 + bx^2 + cx + d$ 定义的曲线的一部分。以下哪一个最小？

(A) P(-1) $P(-1)$

(B) The product of the zeros of P $P$零点积

(D) The sum of the coefficients of P $P$系数和

(E) The sum of the real zeros of P $P$实零点和

Answer

Correct choice: (C)

正确答案：（C）

Solution

Note that there are 3 maxima/minima. Hence we know that the rest of the graph is greater than 10. We approximate each of the above expressions: 1. According to the graph, $P(-1) > 4$ 2. The product of the roots is $d$ by Vieta’s formulas. Also, $d = P(0) > 5$ according to the graph. 3. The product of the real roots is $>5$, and the total product is $<6$ (from above), so the product of the non-real roots is $< \frac{6}{5}$. 4. The sum of the coefficients is $P(1) > 2.5$ 5. The sum of the real roots is $> 5$. Clearly $\mathrm{(C)}$ is the smallest.

注意到图中有 3 个极大/极小点。因此我们知道图像其余部分都大于 10。我们对下面各表达式作近似判断： 1. 由图可知，$P(-1) > 4$ 2. 由韦达定理，根的乘积为 $d$。并且由图可知 $d = P(0) > 5$ 3. 实根的乘积 $>5$，而所有根的乘积 $<6$（由上），所以非实根的乘积 $< \frac{6}{5}$。 4. 系数和为 $P(1) > 2.5$ 5. 实根的和 $> 5$。显然 $\mathrm{(C)}$ 最小。

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