AMC10 2023 B

AMC10 2023 B · Q12

AMC10 2023 B · Q12. It mainly tests Functions basics, Vieta / quadratic relationships (basic).

When the roots of the polynomial \[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot \cdot (x-10)^{10}\] are removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive?

当多项式 \[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot \cdot (x-10)^{10}\] 的根从数轴上移除后，剩下的是$11$个不相交的开区间。这些区间中有多少个上$P(x)$为正？

(A) 3 3

(B) 7 7

(D) 4 4

(E) 5 5

Answer

Correct choice: (C)

正确答案：（C）

Solution

The expressions to the power of even powers are always positive, so we don't need to care about those. We only need to care about $(x-1)^1(x-3)^3(x-5)^5(x-7)^7(x-9)^9$. We need 0, 2, or 4 of the expressions to be negative. The 9 through 10 interval and 10 plus interval make all of the expressions positive. The 5 through 6 and 6 through 7 intervals make two of the expressions negative. The 1 through 2 and 2 through 3 intervals make four of the expressions negative. There are $\boxed{\textbf{(C) 6}}$ intervals.

偶次幂的表达式总是正的，因此我们不需要关心那些。我们只需关心$(x-1)^1(x-3)^3(x-5)^5(x-7)^7(x-9)^9$。我们需要$0$、$2$或$4$个表达式为负。$9$到$10$区间和$10$以上区间使所有表达式为正。$5$到$6$和$6$到$7$区间使两个表达式为负。$1$到$2$和$2$到$3$区间使四个表达式为负。有$\boxed{\textbf{(C) 6}}$个区间。

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