Section: 多元多边函数 | 6.2 多元函数

Section outline

When computer programmers write programs, they ultimately want their programs to run as fast as possible. Some of the fastest programs run in what is called polynomial time. What does polynomial mean? That is what we consider in this section.
::当计算机程序员写入程序时, 他们最终希望他们的程序运行尽可能快。一些最快的程序在所谓的多边时间运行。多国籍程序意味着什么? 这就是我们在本节中考虑的。

What Is a Polynomial Function?
::什么是多面函数?

In previous chapters, you have already considered some polynomial functions—lines (except for vertical lines, which fail the vertical line test). Instead of discussing one group of polynomial functions, we can generalize.
::在前几章中,您已经考虑过一些多面函数线(垂直线线除外,垂直线线线没有通过垂直线测试 ) 。我们不用讨论一组多面函数,而是可以一概而论。

Definition of a Polynomial Function
::多面函数的定义

A function is a polynomial function if it is of the form $\begin{aligned} f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . + a_{2} x^{2} + a_{1} x + a_{0} \end{aligned}$ where the $\begin{aligned} a_{i} \end{aligned}$ are real numbers and the $\begin{aligned} n \end{aligned}$ are non-negative integers .

To recognize a polynomial function, we are looking for terms with real coefficients and the exponents are either positive integers or 0.
::为了确认一个多数值函数,我们正在寻找具有实际系数的条件,指数要么是正整数,要么是0。

Example 1
::例1

Is $\begin{aligned} f (x) = - 4 x^{7} + 5 x^{5} - x^{4} - 9 x^{3} + 2 x - 12 \end{aligned}$ a polynomial function?
::f(x)4x7+5x5-x4-9x3+2x-12是否是一个多圆函数?

Solution: The coefficients are -4, 5, -1 -9, 2, and 12, which are all real numbers. The exponents are 7,5,4,3,1, and 0 ( $\begin{aligned} - 12 x^{0} = - 12 \cdot 1 = - 12 \end{aligned}$ ), which are all positive integers or 0. This is a polynomial function.
::解答: 系数是 4, 5, 1-1-9, 2 和 12, 它们都是真实数字。指数是 7, 5, 4, 3, 1 和 0 ( - 12x0\\\\12) , 它们都是正整数或 0。这是一个多元函数。

Example 2
::例2

Is $\begin{aligned} f (x) = 8 x^{4} + 10 x^{\frac{1}{2}} + 7 \end{aligned}$ a polynomial function?
::f( x) = 8x4+10x12+7 是多圆函数吗 ?

Solution: The coefficients are 8, 10, and 7—all real numbers. However, $\begin{aligned} \frac{1}{2} \end{aligned}$ is not a positive integer, so this is not a polynomial function.
::解答:系数是8、10和7-所有实际数字。然而,12并不是正整数,所以这不是一个多元函数。

Example 3
::例3

Is $\begin{aligned} f (x) = \frac{2}{3} x^{5} - \frac{3}{4} x^{3} + \frac{5}{8} x^{2} \end{aligned}$ a polynomial function?
::f(x) = 23x5-34x3+58x2 是否是一个多元函数?

Solution: The only requirement for the coefficients is that they be real numbers, which these fractions are. 5, 3, and 2 are positive integers, so this is a polynomial function.
::解答:对系数的唯一要求是它们是真实的数字, 这些分数是。 5, 3和2是正整数, 所以这是一个多元函数。

Example 4
::例4

Is $\begin{aligned} f (x) = x^{2} - x^{- 3} + | x | \end{aligned}$ a polynomial function?
::f( x) =x2- x-3 {x} 是否是一个多边函数?

Solution: This is not a polynomial function because the exponent in the second term is a negative integer and the third term includes absolute value bars.
::解决方案 : 这不是一个多数值函数, 因为第二任期的指数为负整数, 第三任期包括绝对值条。

We often refer to by the number of terms in the polynomial. A monomial is a one-term polynomial (the prefix mono- means one, like monorail or a train with one rail). A polynomial with two terms is a binomial (the prefix bi- means two, like bicycle, a cycle with two wheels), and a polynomial with three terms is a trinomial (the prefix tri- means three, like triangle, a three-sided polygon). A polynomial with more than three terms is simply named by its number of terms. For example, a polynomial with five terms is called a five-term polynomial.
::我们经常用多面体中的术语来表示。单面体是一期多面体( 前缀单面体是指一号, 如单轨或一轨列车 ) 。两个条件的多面体是二元体( 前缀双面体是指二号, 如自行车, 两轮) , 三个条件的多面体是三元体( 前缀三号意指三号, 如三角形, 一个三面多边形 ) 。三个条件以上三个条件的多面体只是用其数来表示。例如, 五个条件的多面体体称为五期多面体体。

We also tend to write polynomials in what is called standard form , where we list all the terms in order from highest power to lowest power.
::我们还倾向于以所谓的标准形式写多边纪念书, 列出从最高权力到最低权力的所有术语。

Example 5
::例5

Rewrite $\begin{aligned} x^{3} - 5 x^{2} + 12 x^{4} + 15 - 8 x \end{aligned}$ in standard form.
::在标准格式中重写 x3-5x2+12x4+15-8x。

Solution: To rewrite in standard form, put each term in order, from highest to least power.
::解决方案:以标准格式重写,将每个术语排列顺序,从最高到最低权力。

$\begin{aligned} x^{3} - 5 x^{2} + 12 x^{4} + 15 - 8 x = 12 x^{4} + x^{3} - 5 x^{2} - 8 x + 15 \end{aligned}$

::x3-5x2+12x4+15-8x=12x4+x3-5x2-8x15

Notice the power of each term goes down: 4, then 3, then 2, then 1, and lastly 0.
::注意每个学期的权力下降: 4,3,2,1,最后是0。

by CK-12 demonstrates how to write a polynomial in standard form.
::使用 CK-12 演示如何以标准格式写出多面体。

Important Parts of a Polynomial
::多面体重要部分

The box below highlights some of the parts of a polynomial that are frequently used in applications.
::以下框突出显示在应用中经常使用的多边协议的某些部分。

Parts of a Polynomial
::复数部分的部件

Degree : The highest power of all of the terms in the polynomial.
::学位:多语言中所有术语的最高权力。

Leading term: The term with the highest power in the polynomial.
::领导学期: 多民族主义中拥有最高权力的术语。

Leading coefficient : The coefficient of the leading term.
::主要系数:前一学期系数。

Constant term: The term that is just a number (or the $\begin{aligned} x^{0} \end{aligned}$ -term).
::常数术语: 仅是一个数字( 或 x0- term) 的术语。

Example 5
::例5

Identify the degree, leading term, leading coefficient, and constant term of $\begin{aligned} - 3 x^{4} + 8 x^{6} - 5 x^{3} + 11 x^{2} + 9 x + 12 \end{aligned}$ .
::确定3x4+8x6-5x3+11x2+9x+12的学位、主要术语、主要系数和常数。

Solution: First, we need to write the polynomial in standard form: $\begin{aligned} 8 x^{6} - 3 x^{4} - 5 x^{3} + 11 x^{2} + 9 x + 12 \end{aligned}$ .
::解答:首先,我们需要以标准格式8x6 - 3x4 - 5x3+11x2+9x+12书写多边协议。

The degree is the highest power, which is 6, so this is a degree-6 polynomial.
::学位是最高的权力,是6,所以这是6,多面性。

The leading term is the term that includes the variable to the sixth power: $\begin{aligned} 8 x^{6} \end{aligned}$ .
::前一术语是包含第6位变量的术语:8x6。

The leading coefficient is the coefficient of that term: 8.
::主要系数是该术语的系数:8。

The constant term is the $\begin{aligned} x^{0} \end{aligned}$ -term: 12.
::常数值为x0- 期 : 12 。

by Mathispower4u introduces the basic vocabulary related to polynomials.
::由 Mathispower4u 介绍与多语种有关的基本词汇。

Example 6
::例6

Three computer programs run at the following speeds:
::三个计算机程序以下列速度运行:

$\begin{aligned} Program 1 \to 3 n^{2} + 1000 n \\ Program 2 \to 4 n^{4} + 15 \\ Program 3 \to 6 n + 5000 \end{aligned}$

::方案13n2+1000n 方案24n4+15 方案36n+500

Which program runs the fastest?
::哪个程序运行最快?

Solution: To denote the speed of a program, we consider only the leading terms since they have the highest powers.
::解决方案:为了说明一个方案的速度,我们只考虑主要条件,因为它们拥有最高权力。

$\begin{aligned} Program 1 \to 3 n^{2} \\ Program 2 \to 4 n^{4} \\ Program 3 \to 6 n \end{aligned}$

::方案13n2 方案24n4 方案36n

Then, we remove the leading coefficients, since they do not depend on a variable. What remains is called the order of the function, or more commonly, O ¹ .
::然后,我们去掉主要系数,因为它们不依赖于变量。剩下的是函数的顺序,或者更常见的O1。

">
$\begin{aligned} Program & Leading Term & Order \\ Program 1 & n^{2} & O (n^{2}) \\ Program 2 & n^{4} & O (n^{4}) \\ Program 3 & n & O (n) \end{aligned}$

::1n2O(n2) 方案2n4O(n4) 方案3nO

The fastest program will have the lowest order or power. Program 3 is the fastest since the power of the leading term is 1.
::最快的方案将拥有最低的顺序或权力,方案3是最快的,因为前一学期的权力是1。

Feature: A Bend in the Road
::特色:路中弯弯曲

Civil engineers have a unique job. If it has to do with roads, bridges, canals, dams, or buildings, a civil engineer is usually involved. What do they do exactly? Well, a civil engineer deals with design, construction, and maintenance. They examine the different aspects of whatever is being built and make sure that it is mathematically sound, environmentally sound, and serves the purpose the structure was intended to serve. Civil engineers use polynomials in many aspects of their job.
::土木工程师有独特的工作。如果涉及到道路、桥梁、运河、水坝或建筑,通常会涉及土木工程师。他们究竟要做什么?土木工程师处理设计、建筑和维护。他们检查建筑中的不同方面,确保建筑在数学上是无害的,无害环境的,并符合建筑服务的目的。土木工程师在工作的许多方面使用多面体。

Think about a road. There are very few completely straight roads out there in the world. Most roads must have a curve of some kind. It can be a horizontal curve (side to side), like the one in the first picture, or it can be a vertical curve (a hill or a dip in the road). If you were a civil engineer tasked with designing a road for a particular area, you would have to consider the curves and straight areas of that road. Creating a drawing for construction is not as simple as a sketch. It must show measurements, degrees, angles, and curves, and curves are expressed using polynomials.
::想想道路。世界上很少有一条完全直径的道路。大多数道路必须有某种曲线。它可以是横向曲线( 侧面) , 比如第一幅图中的曲线, 或者可能是垂直曲线( 山坡或者路边的泥巴 ) 。如果您是负责为特定地区设计道路的土木工程师, 您就必须考虑这条道路的曲线和直线区域。为建设绘制一张图不像草图那么简单。它必须显示测量、度、角度、曲线和曲线, 并用多面表示曲线。

Look at the image above. In Atlanta, civil engineers found that if they used a curve in road construction, then they could help cars to travel safely. In this design, the civil engineers used a vertical curve to help automobiles adjust between changes in land elevation. When graphing or designing, data points are connected to form either a straight line or a curve. The degree of the curve is described through a polynomial function. Without an ability to work with polynomials, civil engineers would not be able to accomplish their tasks.
::在亚特兰大,土木工程师发现,如果在道路建设中使用曲线,他们可以帮助汽车安全行驶。在这个设计中,土木工程师使用垂直曲线帮助汽车在陆地高地变化之间进行调整。在图形绘制或设计时,数据点连接成一条直线或曲线。曲线的程度通过一个多面函数来描述。如果没有多面函数的工作能力,土木工程师将无法完成任务。

The first few minutes of the video below demonstrate a problem that can be solved with a quadratic polynomial.
::以下影片的最初几分钟展示了一个可以通过四边多面体解决的问题。

by Daniel Findley demonstrates how the quadratic formula can be used in vertical curve fitting design.
::Daniel Findley展示了二次方程式如何用于垂直曲线安装设计。

Summary
::摘要

A function is a polynomial function if it is the sum or difference of terms with real coefficients and non-negative integer powers.
::函数是一个多数值函数,如果它是与实际系数和非负整数功率之和或条件之差。

Important parts of a polynomial function include the degree, the leading term, the leading coefficient, and the constant term.
::多元函数的重要部分包括学位、主要术语、主要系数和恒定术语。

Review
::回顾

Indicate whether each expression is a polynomial. If it is, e xpress it in standard form and identify the degree, the leading term, the leading coefficient, and the constant term.
::指出每个表达式是否是一个多义表达式。如果是,请以标准格式表示,并标明程度、前一术语、主要系数和常数。

$\begin{aligned} x^{2} + 3 x^{\frac{1}{2}} + 3 \end{aligned}$
::x2+3x12+3

$\begin{aligned} \frac{1}{3} x^{2} y - 9 y^{2} - 4 \end{aligned}$
::13x2y-9y2-4

$\begin{aligned} 3 x^{- 3} \end{aligned}$
::3x-3

$\begin{aligned} \frac{2}{3} t^{2} - \frac{1}{t^{2}} \end{aligned}$
::23t2-1t2

$\begin{aligned} \sqrt{x} - 2 x - 1 \end{aligned}$
::x-2x-1

$\begin{aligned} {(x^{\frac{3}{2}})}^{2} + 5 \end{aligned}$
:x322)2+5

$\begin{aligned} 3 - 2 x \end{aligned}$
::3-2x

$\begin{aligned} \frac{1}{x^{2}} + x + 5 \end{aligned}$
::1x2+x+5

$\begin{aligned} x^{3} + 8 x^{4} - 15 x + 14 x^{2} - 20 \end{aligned}$
::x3+8x4 - 15x+14x2 - 20

$\begin{aligned} x^{3} + 8 \end{aligned}$
::x3+8x3+8

$\begin{aligned} 5 x^{- 2} + 9 x^{- 1} + 16 \end{aligned}$
::5x-2+9x-1+16

$\begin{aligned} x^{2} \sqrt{2} - x \sqrt{6} + 10 \end{aligned}$
::x22 - x6+10 x22 - x6+10

$\begin{aligned} \frac{x^{4} + 8 x^{2} + 12}{3} \end{aligned}$
::x4+8x2+123

$\begin{aligned} \frac{x^{2} - 4}{x} \end{aligned}$
::x2 - 4x

$\begin{aligned} - 6 x^{3} + 7 x^{5} - 10 x^{6} + 19 x^{2} - 3 x + 41 \end{aligned}$
::- 6x3+7x5-10x6+19x2-3x+41

Explore More
::探索更多

1. Write a degree 3 polynomial whose coefficients add up to -8.
::1. 写出第3级多元系数,其系数等于-8。

2. As we saw with lines in slope-intercept form, we can determine the y- intercept of a function by setting $\begin{aligned} x = 0 \end{aligned}$ . Find the y -intercept of the following polynomial functions. How does your result compare to one of the important parts of a polynomial?
::2. 正如我们在斜坡界面的线条中看到的那样,我们可以通过设置 x=0 来确定函数的 Y 界面。找到以下多面函数的 Y 界面。您的结果与多面函数的一个重要部分相比如何?

$\begin{aligned} f (x) = x^{3} + 4 x^{2} - 7 x + 8 & g (x) = 5 x^{4} - 6 x^{2} - 7 & h (x) = 4 x^{6} - x^{3} \end{aligned}$

:x) =x3+4x2 - 7x+8g(x) =5x4 - 6x2 - 7h(x) =4x6 - x3)

3. Which computer program is considered to run the fastest?
::3. 哪些计算机程序被认为运行最快?

$\begin{aligned} Program 1 \to 6 n^{3} + 70 n + 130 \\ Program 2 \to 7 n^{2} + 500 n \\ Program 3 \to 2 n^{4} + 1000 \end{aligned}$

::方案16n3+70n+130 方案27n2+500n

Answers to Review and Explore More Problems
::对审查和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section:
::尝试这一互动,强化本节所探讨的概念:

References
::参考参考资料

1. "Big O Notation," last edited May 16, 2017. .
::1. 2017年5月16日编辑的《大名词》。

Section outline

What Is a Polynomial Function? ::什么是多面函数?

Definition of a Polynomial Function ::多面函数的定义

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Important Parts of a Polynomial ::多面体重要部分

Parts of a Polynomial ::复数部分的部件

Example 5 ::例5

Example 6 ::例6

Feature: A Bend in the Road ::特色:路中弯弯曲

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers to Review and Explore More Problems ::对审查和探讨更多问题的答复

PLIX ::PLIX

References ::参考参考资料

What Is a Polynomial Function?
::什么是多面函数?

Definition of a Polynomial Function
::多面函数的定义

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Important Parts of a Polynomial
::多面体重要部分

Parts of a Polynomial
::复数部分的部件

Example 5
::例5

Example 6
::例6

Feature: A Bend in the Road
::特色:路中弯弯曲

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers to Review and Explore More Problems
::对审查和探讨更多问题的答复

PLIX
::PLIX

References
::参考参考资料