generate a polynomial from the roots -3,2,1,2,3

To generate a polynomial from the given roots, we first need to determine the factors of the polynomial. Since the roots are -3, 2, 1, 2, and 3, the factors of the polynomial are:

(x + 3)(x - 2)(x - 1)(x - 2)(x - 3)

To find the polynomial itself, we multiply these factors:

(x + 3)(x - 2)(x - 1)(x - 2)(x - 3)
= (x^2 + x - 6)(x - 1)(x - 2)(x - 3)
= (x^3 - 3x^2 - 3x + 9)(x - 2)(x - 3)
= (x^4 - 5x^3 + 3x^2 + 47x - 54)

Therefore, the polynomial with roots -3, 2, 1, 2, and 3 is:

f(x) = x^4 - 5x^3 + 3x^2 + 47x - 54

相关问题

How to generate Fitts’ Law to 2D/3D space

### 回答1：
Fitts' Law 是一个在人类-计算机交互中常用的模型，用来预测用户在进行目标选择时所需要的时间。它可以用来估计 2D 和 3D 空间中的指针移动时间。

要生成 Fitts' Law 模型，你需要收集一些用户数据，包括用户在移动指针时所需的时间和所移动的距离。然后，你可以使用下面的公式来计算 Fitts' Law 模型：

MT = a + b * log2(2D/W)

其中：

- MT 是移动时间。
- a 和 b 是常数。
- D 是目标的距离。
- W 是目标的宽度。

你可以使用最小二乘法或其他线性回归方法来拟合这个模型，以便找出最佳的 a 和 b 常数。

### 回答2：
菲茨定律（Fitts' Law）是一种计算人的运动学行为的模型，用于预测人在指向目标并触发动作时的时间和精准度。该定律适用于二维和三维空间。以下是在二维和三维空间中生成菲茨定律的方法：

1. 定义目标：首先，需要明确定义目标，可以是屏幕上的一个按钮、一个物体或一个立体空间中的特定点。

2. 测量任务难度：测量目标的宽度和距离。在二维空间中，目标宽度是指目标的最长边的宽度，距离是指起始点到目标的欧式几何距离。在三维空间中，需要考虑目标的宽度、高度和深度。

3. 计算指数：使用菲茨定律公式计算指数，该公式为：MT = a + b log2(1 + D/W)，其中MT代表指向目标的平均时间，a和b是实验中得到的常数，D表示起始点到目标的距离，W表示目标的宽度。

4. 进行实验：进行一系列实验以收集数据，包括不同距离和不同宽度的目标。在每个实验中记录下参与者移动到目标并触发动作所需的时间。根据实验数据计算得到a和b的值。

5. 分析结果：根据得到的实验数据和计算出的指数，分析目标的难度和准确度。从实验结果中可以得出在不同距离和大小条件下，平均运动时间的变化趋势。

通过以上步骤，可以在二维和三维空间中生成菲茨定律。这个模型可以帮助我们理解人在指向目标并触发动作时的运动学行为，进而优化用户界面和设计，提高用户的交互体验。

### 回答3：
Fitts' Law is a mathematical model that predicts the time required for a user to reach a target based on its size and distance. It is commonly used in human-computer interaction to design efficient user interfaces and input devices. However, applying Fitts' Law to a 2D or 3D space has some unique considerations.

To generate Fitts' Law in a 2D space, we need to define the target size and distance. The target size is typically represented as the width or diameter of the target object, while the distance can be measured as the Euclidean distance between the start and target points. With these measurements, we can calculate the index of difficulty (ID) using the formula:

ID = log2(D/W + 1)

where D is the distance and W is the target width. A higher ID value indicates a greater difficulty in reaching the target.

Next, we can use the index of difficulty to calculate the movement time (MT) using Fitts' Law formula:

MT = a + b * ID

where a and b are experimentally derived constants. The constant a represents the time required for the initial acceleration or deceleration, while b represents the movement speed. By tuning these constants, we can fit the model to real-world data.

When applying Fitts' Law to a 3D space, additional considerations need to be taken into account. Firstly, we need to consider the target's depth or height in addition to its width. The ID calculation would then incorporate both the 2D cursor movement and the 3D depth of the target.

Furthermore, in a 3D space, different techniques and input devices can be utilized, such as 3D cursors or motion controllers. These devices introduce additional factors, such as rotation, orientation, and the interaction techniques employed. These factors need to be accounted for in the model to accurately predict movement time.

In conclusion, to generate Fitts' Law in a 2D/3D space, we need to measure the target size and distance, calculate the index of difficulty, and then predict the movement time using experimentally derived constants. In a 3D space, additional considerations like depth and the chosen interaction techniques need to be factored in.

sql怎么输出1，2，3，4等等

如果你要输出1，2，3，4等等数字序列，可以使用 MySQL 的 `GENERATE_SERIES` 函数。这个函数可以生成指定范围内的数字序列。以下是一个示例 SQL 查询，可以生成从1到10的数字序列：

SELECT generate_series(1, 10);

这将输出以下结果：

 generate_series
-----------------
               1
               2
               3
               4
               5
               6
               7
               8
               9
              10
(10 rows)

你可以根据需要进行调整，例如修改上面的查询，生成从5到15的数字序列：

SELECT generate_series(5, 15);

这将输出以下结果：

 generate_series
-----------------
               5
               6
               7
               8
               9
              10
              11
              12
              13
              14
              15
(11 rows)