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多元结构和自论结构
The Purpose of this Lesson
::本课程的目的

In this lesson, you will explore polynomial expressions by comparing and contrasting them with integers . You'll learn about their structure and how to perform arithmetic operations with them.
::在此课中, 您将会通过比较和对比它们与整数来探索多元表达式。您将会了解它们的结构以及如何使用它们进行算术操作。

Increasing variable exponents depicted as increasing numbers of dimensions
::越来越多的可变指数,被描述为数量不断增多的维度

Activity 1: Distinguishing Function Types
::活动1:区分职能类型

Example 1-1
::例1-1

In the previous chapter, you were introduced to many different types of functions. I dentify the function types by name and discuss their characteristics. Write the general form for each.
$\begin{align*}\begin{array}{ll} \quad \text{a}. & y=\frac23 x +7\\ \quad \text{b.} & y=8x^2\\ \quad \text{c.} & y=\frac{6}{x}\\ \quad \text{d.} & y=300(\frac12)^x\\ \quad \text{e.} & y=x\\ \quad \text{f.} & y=\frac{12}{x^2}\\ \quad \text{g.} & y=\left(\sqrt3\right)x+\sqrt{10}\\ \quad \text{h.} & y=\sqrt{x}\\ \end{array} \end{align*}$
::在前一章中, 您被引入了多种不同的函数类型。按名称识别函数类型, 并讨论其特性。写入每个函数的一般格式。 a. y= 23x+7b.y= 8x2c.y= 6xd.y= 300( 12) xe.y=xf.y= 12x2g.y= (3)x+10h.y=x

Solution:
$\begin{aligned} \begin{array}{lllll} Function & Type & General Form \\ a . & y = \frac{2}{3} x + 7 & Linear & y = m x + b \\ b. & y = 8 x^{2} & Quadratic & y = a x^{2} \\ c. & y = \frac{6}{x} & Rational & y = \frac{a}{x} \\ d. & y = 300 (\frac{1}{2})^{x} & Exponential & y = a (b)^{x} \\ e. & y = x & Linear & y = m x + b \\ f. & y = \frac{12}{x^{2}} & Rational & y = \frac{a}{x^{2}} \\ g. & y = (\sqrt{3}) x + \sqrt{10} & Linear & y = m x + b \\ h. & y = \sqrt{x} & Square Root & y = a \sqrt{x} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{ll|l|l|l} & \text{Function} & \text{Type} & \text{General Form} \\ \hline \text{a}. & y=\frac23 x +7 & \text{Linear} & y=mx+b\\ \text{b.} & y=8x^2 & \text{Quadratic} & y=ax^2\\ \text{c.} & y=\frac{6}{x} & \text{Rational} & y=\frac{a}{x}\\ \text{d.} & y=300(\frac12)^x & \text{Exponential} & y=a(b)^x\\ \text{e.} & y=x & \text{Linear} & y=mx+b\\ \text{f.} & y=\frac{12}{x^2} & \text{Rational} & y=\frac{a}{x^2}\\ \text{g.} & y=\left(\sqrt3\right)x+\sqrt{10} & \text{Linear} & y=mx+b\\ \text{h.} & y=\sqrt{x} & \text{Square Root} & y=a\sqrt{x}\\ \end{array} \end{align*}$
::解析度: Forma.y = 23x+7Lineary= mx+bb.y = 8x2Quadraticy= 轴2c.y = 6xRationaly=axxaxd.y= 300(12)xExpentialy=a(b)xxe.y=xLineary=mx+bf.y= 12x2Rationaly=ax2g.y=(3)x+10Lineary=mx+bh.y=xSquarerooty=ax

Example 1-2

::例1-2

A polynomial is a broader type of function . In fact, four of the functions above are . The functions above are sorted below. What similarities and differences do you see between the polynomials ? How are they different from the non-polynomials?

$\begin{aligned} \begin{array}{ll} Polynomials & Non-polynomials \\ y = \frac{2}{3} x + 7 & y = \frac{6}{x} \\ y = 8 x^{2} & y = 300 (\frac{1}{2})^{x} \\ y = x & y = \frac{12}{x^{2}} \\ y = (\sqrt{3}) x + \sqrt{10} & y = \sqrt{x} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{l|l} \text{Polynomials} & \text{Non-polynomials}\\ \hline y=\frac23 x +7 & y=\frac{6}{x}\\ y=8x^2 & y=300(\frac12)^x\\ y=x & y=\frac{12}{x^2}\\ y=\left( \sqrt3 \right) x+\sqrt{10} & y=\sqrt{x}\\ \end{array}\end{align*}$
::多多边性是一个更广泛的函数类型。事实上, 以上四个函数是。以上四个函数在下面排序。以上函数在下面排序。您看到多多边性之间有什么相似和不同之处? 它们与非多球性有什么不同? 它们与非多球性多球性=23x+7y=6xy=8x2y=300( 12xy=xy=12x2y=(3)x+10y=x

Solution: Each polynomial in the list features a real number times a variable or a variable squared. Sometimes a constant is added. The variable is not in the denominator of a fraction . The variable is not in the exponential position. The variable is not square rooted.
::解析 : 列表中的每个多数值都含有一个变量或一个变量方形的实际数字乘以一个变量或变量方形。有时会添加一个常数。该变量不在分数的分母中。该变量不在指数位置中。该变量不是正方根。

Work It Out
::F. 工作外外

Categorize each of the following as polynomials or non-polynomials. Explain your choices.
$\begin{align*}\begin{array}{ll} \quad \text{a}. & y=-5.8x +\pi\\ \quad \text{b.} & y=\frac{x}{7}\\ \quad \text{c.} & y=-\frac{x}{\sqrt2}+12\\ \quad \text{d.} & y=\pi(4)^x\\ \quad \text{e.} & y=\pi x-\sqrt3\\ \quad \text{f.} & y=\frac{1}{x}\\ \quad \text{g.} & y=\frac{x}{1}\\ \quad \text{h.} & y=\pi x^2\\ \end{array}\end{align*}$
::以下每一种分类为多球或非球类。请解释您选择的选项。 a. y5. 8xb. y=x7c. y\\ x2+12d. y\\ (4)xe. yx-3f.y=1xg.y=x1h. y\\x2

The sorting exercise in the last problem was complicated by the presence of . List all the irrational numbers used in the last problem. List all the rational numbers used in the last problem.
::最后一个问题中的排序操作因存在 . 而变得复杂。列出最后一个问题中使用的所有非理性数字。列出最后一个问题中使用的所有合理数字。

Activity 2 : Integers and Polynomials

::活动2:整数和聚合体

Polynomial expressions are very similar to integers in terms of their structure. In this and subsequent examples, we will explore the structure of integers and relate it to the structure of polynomials.
::多元表达式的结构与整数结构非常相似。在此及随后的示例中, 我们将探索整数结构, 并将其与多元结构联系起来。

Example 2-1
::例2-1

Consider the number 743. It is composed of 3 digits. What does each digit represent? Write 743 as the sum of 3 numbers. Then write it as the sum of 3 numbers in scientific notation . Write several integers of your choice and write them as sums as you did for 743.
::考虑数字 743 。它由 3 位数组成。每个位数代表什么 ? 将 743 写为 3 位数之和。然后在科学符号中将 743 写为 3 位数之和。写几个您选择的整数, 并按您对 743 的整数写成 743 。

Solution:
$\begin{aligned} \begin{array}{lll} Digit & What it Represents & Scientific Notation \\ 7 & 700 & 7 \cdot 10^{2} \\ 4 & 40 & 4 \cdot 10^{1} \\ 3 & 3 & 3 \cdot 10^{0} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{l|l|l} \text{Digit} & \text{What it Represents} & \text{Scientific Notation}\\ \hline 7 & 700 & 7\cdot10^2\\ 4 & 40 & 4\cdot10^1\\ 3 & 3 & 3\cdot 10^0\\ \end{array}\end{align*}$

$\begin{aligned} 743 = 7 (10)^{2} + 4 (10)^{1} + 3 (10)^{0} \end{aligned}$
$\begin{align*}743=7(10)^2+4(10)^1+3(10)^0\end{align*}$

Example 2-2
::例2-2

Polynomials have the same structure as the integer written above. Each of the following is a polynomial. Write a number that is similar to each polynomial. A suggested answer is given for the first one .

$\begin{aligned} \begin{array}{llll} Polynomial & Number & Simplified & Simplified \\ Number & Polynomial \\ 4 x^{1} + 5 x^{0} & 4 (10)^{1} + 5 (10)^{0} & 45 & 4 x + 5 \\ 3 x^{2} + 2 x^{1} + 6 x^{0} \\ 5 x^{3} + x^{2} + 7 x^{1} + 0 x^{0} \\ 12 x^{4} + 7 x^{3} + 3 x^{2} + 0 x^{1} + 1 x^{0} \\ 3 x^{0} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{l|l|l|l} \text{Polynomial} & \text{Number} & \text{Simplified} & \text{Simplified}\\ \ & \ & \text{ Number} & \text{Polynomial}\\ \hline 4x^1+5x^0 & 4(10)^1+5(10)^0 & 45 & 4x+5\\ 3x^2+2x^1+6x^0 & &&\\ 5x^3+x^2+7x^1+0x^0&&&\\ 12x^4+7x^3+3x^2+0x^1+1x^0&&&\\ 3x^0 &&&\\ \end{array}\end{align*}$
::与上面写入的整数结构相同。以下各为多数值。写一个与每个多数值相似的数。给出第一个给出建议的答复。简单化的多数值。简单化的多球数 : Polynomomial4x1+5x04 (1010) 1+5 (10) 00454x+53x2+53x2+2x1+6x05x1+6x5x3+x2+7x1+0x012x4+7x3+3x2+0x1+1x03x0

Solution:
::解决方案 :

$\begin{aligned} \begin{array}{llll} Polynomial & Number & Simplified & Simplified \\ Number & Polynomial \\ 4 x^{1} + 5 x^{0} & 4 (10)^{1} + 5 (10)^{0} & 45 & 4 x + 5 \\ 3 x^{2} + 2 x^{1} + 6 x^{0} & 3 (10)^{2} + 2 (10)^{1} + 6 (10)^{0} & 326 & 3 x^{2} + 2 x + 6 \\ 5 x^{3} + 1 x^{2} + 7 x^{1} + 0 x^{0} & 5 (10)^{3} + 1 (10)^{2} + 7 (10)^{1} + 0 & 5170 & 5 x^{3} + x^{2} + 7 x \\ 3 x^{0} & 3 (10)^{0} & 3 & 3 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{l|l|l|l} \text{Polynomial} & \text{Number} & \text{Simplified} & \text{Simplified}\\ \ & \ & \ \text{ Number} & \text{Polynomial}\\ \hline 4x^1+5x^0 & 4(10)^1+5(10)^0 & 45 & 4x+5\\ 3x^2+2x^1+6x^0 & 3(10)^2+2(10)^1+6(10)^0 & 326 & 3x^2+2x+6\\ 5x^3+1x^2+7x^1+0x^0 & 5(10)^3+1(10)^2+7(10)^1+0&5170& 5x^3+x^2+7x\\ 3x^0 & 3(10)^0 & 3 & 3\\ \end{array}\end{align*}$
::简单化数( Polynomomial4x1+5x04( 1010) 1+5 (10) 0454x+53x2+2x1+6x03 (1010) 2+2( 102+2( 10) 1+6 (1003263x2+2x) 1+65263x2+2x2x+653+1x3+1x1+7x1+0x05( 1010)3+12( 10) 2+7( 10) 1+1+051705x3+x2+7x3x03( 1010) 033

The Structure of Polynomials
::一夫多妻制结构

Polynomials consist of terms connected by addition or subtraction .
::多面体由通过增减相连接的术语组成。

Each term consists of a real number called a coefficient , times a variable, raised to a power.
::每个术语由实际数字组成,称为系数,乘变量,升至权力。

Each power is a natural number (an integer greater than or equal to zero).
::每个功率都是自然数(整数大于或等于零)。

Work it Out
::工作出来

Create your own integers to experiment and answer the following questions. Is the sum of two integers always an integer? Is the difference between two integers always an integer? Is the product of two integers always an integer? Is the quotient of two integers always an integer? Explain your reasoning.
::创建您自己的整数来实验并回答下面的问题。两个整数的总和总是整数吗? 两个整数的差数总是整数吗? 两个整数的产物总是整数吗? 两个整数的商数总是整数吗? 解释你的推理。

Create your own polynomials to experiment and answer the following questions. Is the sum of two polynomials always a polynomial? Is the difference between two polynomials always a polynomial? Explain your reasoning. How is this similar to what you saw with integers in the last problem?
::创建您自己的多面体来实验和回答下列问题。两个多面体的总和总是多面体吗? 两个多面体的区别总是多面体吗? 解释一下你的推理。这与您在最后一个问题中看到的整数相似吗 ?

(extension) Products and quotients of two polynomials were not explored in the last problem. Is the product of two polynomials always a polynomial? Is the quotient of two polynomials always a polynomial? Explain your reasoning. How is this similar to your results with integers in number 8?
::最后一个问题没有探讨两个多面体的产品和商数。两个多面体的产物总是多面体吗? 两个多面体的产物总是多面体吗? 两个多面体的商数总是多面体吗? 解释一下你的推理。这和8个整数的结果有什么相似之处?

PLIX Interactive
::PLIX 交互式互动

Arithmetic with Polynomials
::对多面体进行自解

Polynomials form a closed set under addition, subtraction, and multiplication . That means that adding, subtracting, or multiplying two polynomials results in a polynomial.
::多面体形成一个封闭的集合,在添加、减法和乘法下。这意味着增加、减法或乘法两个多面体导致多面体。

Integers are also closed under addition, subtraction, and multiplication.
::在加法、减法和乘法下也关闭整数。

Polynomials and integers are not closed under division . Dividing integers results in a rational number . results in a rational expression .
::单数和整数不按除法闭合。分数整数产生合理数字。结果产生合理表达式。

Activity 3 : Polynomial Structure and Arithmetic
::活动3:多面结构和再分析

Interactive
::交互式互动

In the interactive below, move the red point to explore polynomials of varying degree . Discuss how the degree is related to each polynomial and find the number of terms each polynomial has.
::在下文互动部分, 移动红点以探索不同程度的多元性。讨论该程度与每个多元性的关系, 并找到每个多元性的术语数量。

Work it Out
::工作出来

The linear and quadratic functions you already worked with are a subset of the polynomials. You've already compared and contrasted linear and quadratic functions. In terms of their equations, what is the most important difference between linear and quadratic functions? Why? Give an example of each to support your argument.
::您已经工作过的线性和二次函数是多式函数的一个子集。您已经比较和对比了线性和二次函数。从它们的方程来看, 线性和二次函数之间最重要的区别是什么? 为什么? 请举一个例子来支持您的论点。

There are two ways of classifying polynomials. First, you can classify polynomials by the largest exponent found. What is the largest exponent of a quadratic expression ? What is the largest exponent of a linear expression ? The largest exponent is called the degree of the polynomial. State the degree of each of the following polynomials:
$\begin{align*}\begin{array}{ll} \quad \text{a.}& 10x^2\\ \quad \text{b.}& 5x-12\\ \quad \text{c.}& -6x^3+2x^2-x+9\\ \quad \text{d.}& x^{100}+x^{99}-11\\ \quad \text{e.}& 456x^7+15x^6-4x^5+2x^2-1\\ \quad \text{f.}& x^{10}-x^{91}+x^5+x^{122}+x^3\\ \end{array} \end{align*}$
::有两种方法可以分类多面性。首先, 您可以用发现的最大前端对多面性进行分类。二次表达式的最大引号是什么? 线性表达式的最大引号是什么? 什么是线性表达式的最大引号? 最大的引号叫多面性的程度。请指出以下多种多面性的程度: a.10x2b.5x- 12c.- 6x3+2x2x2- x- x+9d. x100+x9911- e.4567+15x6- 4x5+2x2- 1f. x10- x91+x5+x122+x3

The last polynomial in the last problem was trickier than the rest. Why? What do you think is the most logical way to organize a polynomial? When polynomials are organized from term of greatest degree to term of least degree, the polynomial is in standard form . Re-write the last polynomial in standard form.
::最后一个问题中最后一个多语种比其他问题更狡猾。为什么? 您认为什么是组织多语种的最合乎逻辑的方法? 当多语种从最大到最低的术语组织起来时, 多语种以标准形式出现。以标准形式重新写入最后一个多语种形式。

We can also classify polynomials by the number of their terms. If a polynomial has one term, it's a monomial . If it has two terms, it's a binomial , and if three, a trinomial . Beyond three you just state the number of terms the polynomial has. Create your own monomial, binomial, and trinomial. Create a trinomial of degree 5. Create a binomial of degree 7. Create a monomial of degree 0.
::我们还可以按照多式术语的编号来分类多式术语。如果多式术语有一个术语, 它是一个单式术语。如果它有两个术语, 它是一个二式术语, 如果三个术语, 一个三式术语。在三个多式术语之后, 您只需说明多式术语的编号。创建您自己的单式、二式和三式术语。创建一个等级的三式变量 5 。创建一个等级为 7 的二进制变量。创建一个等级为 0 的单一名称。

Is it possible to create a trinomial of degree 1? Why or why not? Is it possible to create a binomial of degree 0? Why or why not? What happens when the coefficient of a term is 0? Explain.
::是否有可能创造第1级的三进制? 为什么不行? 是否有可能创造第0级的二进制?为什么不行?当一个学期的系数是0时会发生什么?解释一下。

The expression $\begin{align*}3x+4x-17\end{align*}$ looks like a trinomial of degree 1. Is this expression completely simplified? Why or why not? Simplify it further if possible. How would you classify this polynomial, in terms of degree and number of terms?
::3x+4x- 17 表达式看起来像是度的三进制 1: 这个表达式是否完全简化了? 为什么? 如果可能的话, 进一步简化它。您将如何从程度和数量上将这个多义分类 ?

The following polynomial expressions are not simplified. Combine like terms and write the result in standard form. Then identify each according to degree and number of terms. First, an example is completed for you:
::以下多语种表达式不简化。将类似术语合并, 并将结果写入标准格式。然后根据语句的程度和数量, 确定每个词。首先, 您将完成一个示例 :

$\begin{aligned} \begin{array}{ll} Expression & Explanation \\ 8 + 3 x^{7} + 6 x^{2} - 2 x + 2 x^{7} - 4 x^{2} + 3 & Given polynomial \\ 3 x^{7} + 2 x^{7} + 6 x^{2} - 4 x^{2} - 2 x + 8 + 3 & Re-arranged so that like terms are together \\ 5 x^{7} + 2 x^{2} - 2 x + 11 & Combined like terms \\ Confirm it's written in standard form. \\ Degree is 7 \\ There are 4 terms. \end{array} \end{aligned}$
$\begin{align*}\begin{array}{l|l} \text{Expression} &\text{Explanation}\\ \hline 8+3x^7+6x^2-2x+2x^7-4x^2+3 & \text{Given polynomial}\\ 3x^7+2x^7+6x^2-4x^2-2x+8+3 & \text{Re-arranged so that like terms are together}\\ 5x^7+2x^2-2x+11 & \text{Combined like terms}\\ & \text{Confirm it's written in standard form.}\\ & \text{Degree is}~7\\ & \text{There are}~4~\text{terms.} \end{array} \end{align*}$
::表达式Explaination8+3x7+6x7+6x2+2x2+2x7-4x2+6x7+2x7+6x2_4x2+2x2+2+8+3重新排列,以便类似术语的集合5x7+2x2-2x+11组合与术语一样确认其以标准格式写成。

$\begin{align*}\begin{array}{ll} \quad \text{a.}&5x^2-6x^2+7x-7x+3-14\\ \quad \text{b.}&-1+3x^2+4x^3-6x^5+1\\ \quad \text{c}.&x^2+2x^2+3x^2+4x^2+5x^5-10x^2+12\\ \quad \text{d.}&\frac13 x^2+\frac34 x^2-\frac25x^3-\frac12x^3\\ \end{array} \end{align*}$
::a.5x2 - 6x2+7x7x+7x+7x+3-3-14b.-1+3x2+4x3-6x5+1c.x2+2x2+3x2+3x2+4x2+5x5-10x2+12d.13x2+34x2-225x3-12x3

Summary
::摘要

You can classify polynomials by degree. The degree is the largest exponent. Expressions of degree 1 are linear. Expressions of degree 2 are quadratic.
::您可以按度对多数值进行分类。学位是最大的指数。第1级的表达式是线性的。第2级的表达式是二次的。

You can classify polynomials by the number of terms. Polynomials with 1, 2, or 3 terms are called monomials, binomials, and trinomials, respectively.
::您可以按术语编号对多义进行分类。一、二、三等的多义分别称为单名、二等义和三等义。

Standard form for a polynomial means the polynomial is simplified and the terms are written in descending order of their degree.
::多边式标准格式是指简化多边式标准格式,术语按其程度的降序写成。

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