Course: 9.1 标准表格中的多面体

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section 标准表格中的多面体

Collapse Expand
标准表格中的多面体
Polynomials in Standard Form
::标准表格中的多面体

So far we’ve seen functions described by straight lines (linear functions) and functions where the variable appeared in the exponent (exponential functions). In this section we’ll introduce polynomial functions. A polynomial is made up of different terms that contain positive integer powers of the variables. Here is an example of a polynomial:
::到目前为止,我们已经看到直线函数(线性函数)描述的函数,以及变量出现在引号中的函数(Exponicial 函数)。在本节中,我们将引入多元函数。多元函数由包含变量正整数功率的不同术语组成。这里是多元函数的例子:

$\begin{aligned} 4 x^{3} + 2 x^{2} - 3 x + 1 \end{aligned}$

::4x3+2x2 - 3x+1

Each part of the polynomial that is added or subtracted is called a term of the polynomial. The example above is a polynomial with four terms .
::添加或减去的多元性的每一部分都称为多元性术语。上面的例子是一个包含四个术语的多元性术语。

The numbers appearing in each term in front of the variable are called the coefficients . The number appearing all by itself without a variable is called a constant .
::在变量前面的每个术语中出现的数字被称为系数。没有变量的数值本身就被称作常数。

In this case the coefficient of $\begin{aligned} x^{3} \end{aligned}$ is 4 , the coefficient of $\begin{aligned} x^{2} \end{aligned}$ is 2 , the coefficient of $\begin{aligned} x \end{aligned}$ is -3 and the constant is 1 .
::在这种情况下,x3系数为4,x2系数为2,x系数为-3,常数为1。

Degrees of and Standard Form
::程度和标准表格

Each term in the polynomial has a different degree . The degree of the term is the power of the variable in that term.
::多义中每个词的等级不同。该词的等级是该词中变量的功率。

$\begin{aligned} 4 x^{3} & has degree 3 and is called a cubic term or 3^{r d} order term . \\ 2 x^{2} & has degree 2 and is called a quadratic term or 2^{n d} order term . \\ - 3 x & has degree 1 and is called a linear term or 1^{s t} order term . \\ 1 & has degree 0 and is called the constant . \end{aligned}$

::4x3has 度 3, 被称为立方任期或第3级术语 2x2has 度 2, 被称为二次术语或第2级术语。 - 3xhas 度 1, 称为线性术语或第1级术语。 2x2has 度 0, 称为常数。

By definition, the degree of the polynomial is the same as the degree of the term with the highest degree. This example is a polynomial of degree 3, which is also called a “cubic” polynomial. (Why do you think it is called a cubic?).
::从定义上看,多元度的程度与具有最高程度的术语的程度相同。这个例子就是3级的多元度,也称为“立方”多元度。 (你为什么认为它被称为立方体? )

Polynomials can have more than one variable. Here is another example of a polynomial:
::多面体可以有一个以上的变量。下面是多面体的另一个示例:

$\begin{aligned} t^{4} - 6 s^{3} t^{2} - 12 s t + 4 s^{4} - 5 \end{aligned}$

::t4 - 6s3t2 - 12+4s4 - 5

This is a polynomial because all the exponents on the variables are positive integers. This polynomial has five terms. Let’s look at each term more closely.
::这是一个多数值, 因为变量上的所有引言都是正整数。这个多数值有五个条件。让我们仔细看一看每个词。

Note: The degree of a term is the sum of the powers on each variable in the term. In other words, the degree of each term is the number of variables that are multiplied together in that term, whether those variables are the same or different.
::注:一个术语的程度是该术语中每个变量的功率总和。换句话说,每个术语的程度是该术语中相乘的变量数量,无论这些变量是相同的还是不同的。

$\begin{aligned} t^{4} & has a degree of 4, so it's a 4^{t h} order term \\ - 6 s^{3} t^{2} & has a degree of 5, so it's a 5^{t h} order term . \\ - 12 s t & has a degree of 2, so it's a 2^{n d} order term . \\ 4 s^{4} & has a degree of 4, so it's a 4^{t h} order term . \\ - 5 & is a constant, so its degree is 0. \end{aligned}$

::4级,4级,4级,4级,5级,5级,5级。 12级,2级,2级,4级,4级,4级,5级,5级,5级,0级。

Since the highest degree of a term in this polynomial is 5, then this is polynomial of degree $\begin{aligned} 5^{t h} \end{aligned}$ or a $\begin{aligned} 5^{t h} \end{aligned}$ order polynomial.
::由于这个多民族学系中一个学期的最高学位是5, 那么这是第五或第五级多民族学系。

A polynomial that has only one term has a special name. It is called a monomial ( mono means one). A monomial can be a constant, a variable, or a product of a constant and one or more variables. You can see that each term in a polynomial is a monomial, so a polynomial is just the sum of several monomials. Here are some examples of monomials:
::只有一个术语的单名是一个特殊名称。它被称为单名( 单名意指一个)。单名可以是常数、变量或常数及一个或一个以上变量的产物。您可以看到, 多名词中每个术语都是单名, 所以多名就是数个单名的总和。以下是一些单名的例子 :

$\begin{aligned} b^{2} - 2 a b^{2} 8 \frac{1}{4} x^{4} - 29 x y \end{aligned}$

::b2-2ab2814x4-29xy

Identifying Constants and the Degree of a Polynomial
::识别常数和多元度

For the following polynomials, identify the coefficient of each term, the constant, the degree of each term and the degree of the polynomial.
::对于以下多语种,请标明每个术语的系数、常数、每个术语的程度和多语种的程度。

a) $\begin{aligned} x^{5} - 3 x^{3} + 4 x^{2} - 5 x + 7 \end{aligned}$
::a) x5-3x3+4x2-5x+7

$\begin{aligned} x^{5} - 3 x^{3} + 4 x^{2} - 5 x + 7 \end{aligned}$
::x5 - 3x3+4x2 - 5x+7

The coefficients of each term in order are 1, -3, 4, and -5 and the constant is 7.
::每个术语的系数依次为1、3、4和-5,常数为7。

The degrees of each term are 5, 3, 2, 1, and 0. Therefore the degree of the polynomial is 5.
::每个学期的学位为5、3、2、1和0。因此,多学制的学位为5。

b) $\begin{aligned} x^{4} - 3 x^{3} y^{2} + 8 x - 12 \end{aligned}$
::b) x4-3x3y2+8x-12

$\begin{aligned} x^{4} - 3 x^{3} y^{2} + 8 x - 12 \end{aligned}$
::x4 - 3x3y2+8x- 12

The coefficients of each term in order are 1, -3, and 8 and the constant is -12.
::每个学期的系数依次为1-3和8,常数为-12。

The degrees of each term are 4, 5, 1, and 0. Therefore the degree of the polynomial is 5.
::每个学期的学位为4、5、1和0,因此,多学制的学位为5。

Identifying Polynomials
::识别多面体

Identify the following expressions as polynomials or non-polynomials.
::表示下列表达式为多边或非多边表达式。

a) $\begin{aligned} 5 x^{5} - 2 x \end{aligned}$
:a) 5x5-2x

This is a polynomial.
::这是多面性。

b) $\begin{aligned} 3 x^{2} - 2 x^{- 2} \end{aligned}$
:b) 3x2-2-2-2

This is not a polynomial because it has a negative exponent.
::这不是一个多元性的,因为它有一个消极的推论。

c) $\begin{aligned} x \sqrt{x} - 1 \end{aligned}$
:c)xx-1

This is not a polynomial because it has a radical .
::这不是一个多元的,因为它有激进的。

d) $\begin{aligned} \frac{5}{x^{3} + 1} \end{aligned}$
:d) 5x3+1

This is not a polynomial because the power of $\begin{aligned} x \end{aligned}$ appears in the denominator of a fraction (and there is no way to rewrite it so that it does not).
::这不是一个多义, 因为 x 的力量出现在分数的分母中( 并且没有办法重写它, 以免它重写 ) 。

e) $\begin{aligned} 4 x^{\frac{1}{3}} \end{aligned}$
::e) 4x13

This is not a polynomial because it has a fractional exponent.
::这不是一个多元性的,因为它有一个分数的表率。

f) $\begin{aligned} 4 x y^{2} - 2 x^{2} y - 3 + y^{3} - 3 x^{3} \end{aligned}$
:f) 4xy2-2x2y-3+y3-3x3 4xy2-2x2y-3+y3-3x3

This is a polynomial.
::这是多面性。

Often, we arrange the terms in a polynomial in order of decreasing power. This is called standard form .
::通常,我们按照减速的顺序, 以多面形排列术语。这被称为标准形式。

The following polynomials are in standard form:
::下列多面体为标准形式:

$\begin{aligned} 4 x^{4} - 3 x^{3} + 2 x^{2} - x + 1 \end{aligned}$

::4x4 - 3x3+2x2 - x+1

$\begin{aligned} a^{4} b^{3} - 2 a^{3} b^{3} + 3 a^{4} b - 5 a b^{2} + 2 \end{aligned}$

::a4b3-2a3b3+3a4b-5ab2+2

The first term of a polynomial in standard form is called the leading term , and the coefficient of the leading term is called the leading coefficient .
::标准形式的多民族制的第一个任期称为前一任期,前一任期的系数称为前一系数。

The first polynomial above has the leading term $\begin{aligned} 4 x^{4}, \end{aligned}$ and the leading coefficient is 4.
::以上第一个多元系数为4x4,主要系数为4。

The second polynomial above has the leading term $\begin{aligned} a^{4} b^{3}, \end{aligned}$ and the leading coefficient is 1.
::以上第二个多元系数为a4b3,前一术语,主要系数为1。

Writing Polynomials in Standard Form
::在标准表格中写入多面体

Rearrange the terms in the following polynomials so that they are in standard form. Indicate the leading term and leading coefficient of each polynomial.
::将术语重新排列为以下多面体,使其以标准形式出现。指出每个多面体的前一术语和主要系数。

a) $\begin{aligned} 7 - 3 x^{3} + 4 x \end{aligned}$
:a) 7-3x3+4x

$\begin{aligned} 7 - 3 x^{3} + 4 x \end{aligned}$ becomes $\begin{aligned} - 3 x^{3} + 4 x + 7 \end{aligned}$ . Leading term is $\begin{aligned} - 3 x^{3} \end{aligned}$ ; leading coefficient is -3.
::7-3x3+4x成为-3x3+4x+7. 领先学期为-3x3;主要系数为-3。

b) $\begin{aligned} a b - a^{3} + 2 b \end{aligned}$
:b) ab-a3+2b

$\begin{aligned} a b - a^{3} + 2 b \end{aligned}$ becomes $\begin{aligned} - a^{3} + a b + 2 b \end{aligned}$ . Leading term is $\begin{aligned} - a^{3} \end{aligned}$ ; leading coefficient is -1.
::ab-a3+2b 成为-a3+ab+2b。领先术语为-a3;主要系数为-1。

c) $\begin{aligned} - 4 b + 4 + b^{2} \end{aligned}$
:c) - 4b+4+b2

$\begin{aligned} - 4 b + 4 + b^{2} \end{aligned}$ becomes $\begin{aligned} b^{2} - 4 b + 4 \end{aligned}$ . Leading term is $\begin{aligned} b^{2} \end{aligned}$ ; leading coefficient is 1.
::-4b+4+b2成为b2-4b+4,主要术语为b2;主要系数为1。

Simplifying Polynomials
::简化多面体

A polynomial is simplified if it has no terms that are alike. Like terms are terms in the polynomial that have the same variable(s) with the same exponents, whether they have the same or different coefficients.
::多元性如果没有相似的术语,则会简化。类似术语一样,多边性术语的术语具有相同的变量,具有相同的指数,无论它们具有相同的或不同的系数。

For example, $\begin{aligned} 2 x^{2} y \end{aligned}$ and $\begin{aligned} 5 x^{2} y \end{aligned}$ are like terms, but $\begin{aligned} 6 x^{2} y \end{aligned}$ and $\begin{aligned} 6 x y^{2} \end{aligned}$ are not like terms.
::例如,2x2y和5x2y相似,但6x2y和6xy2则不同。

When a polynomial has like terms, we can simplify it by combining those terms.
::当一个多民族的术语相似时, 我们可以通过合并这些术语来简化它。

$\begin{aligned} x^{2} + \underline{6 x y} - \underline{4 x y} + y^{2} \\ ↗ ↖ \\ Like terms \end{aligned}$

::x2+6xy @4xy @ y2 @ 类似条件

We can simplify this polynomial by combining the like terms $\begin{aligned} 6 x y \end{aligned}$ and $\begin{aligned} - 4 x y \end{aligned}$ into $\begin{aligned} (6 - 4) x y \end{aligned}$ , or $\begin{aligned} 2 x y \end{aligned}$ . The new polynomial is $\begin{aligned} x^{2} + 2 x y + y^{2} \end{aligned}$ .
::我们可以将类似 6xy 和 4xy 等词合并为 6-4 xy 或 2xy , 从而简化这个多数值。新的多数值为 x2+2xy+y2 。

Simplify the following polynomials by collecting like terms and combining them.
::通过收集类似术语并将其合并,简化以下多义词集。

a) $\begin{aligned} 2 x - 4 x^{2} + 6 + x^{2} - 4 + 4 x \end{aligned}$
:a) 2x-4x2+6+x2-4+4x

Rearrange the terms so that like terms are grouped together: $\begin{aligned} (- 4 x^{2} + x^{2}) + (2 x + 4 x) + (6 - 4) \end{aligned}$
::重新排列术语,使类似术语的术语组合为- 4x2+x2)+(-2x+4x)+( 6- 4)

Combine each set of like terms: $\begin{aligned} - 3 x^{2} + 6 x + 2 \end{aligned}$
::组合每组类似条件 : - 3x2+6x+2

b) $\begin{aligned} 2 x - 4 x^{2} + 6 + x^{2} - 4 + 4 x \end{aligned}$
::b) 2x-4x2+6+x2-4+4x

Rearrange the terms so that like terms are grouped together: $\begin{aligned} (a^{3} b^{3} - a^{3} b^{3}) + (- 5 a b^{4} + 3 a b^{4}) + 2 a^{3} b - a^{2} b \end{aligned}$
::重新排列术语,将类似术语分组a3b3-a3b3)+(-5ab4+3ab4)+2a3b-a2b)

Combine each set of like terms: $\begin{aligned} 0 - 2 a b^{4} + 2 a^{3} b - a^{2} b = - 2 a b^{4} + 2 a^{3} b - a^{2} b \end{aligned}$
::每组类似条件组合: 0-2ab4+2a3b-a2b2a4+2a3b-a2b2a4+2a3b-a2b

Example
::示例示例示例示例

Example 1
::例1

Simplify and rewrite the following polynomial in standard form. State the degree of the polynomial.
::以标准格式简化和重写以下多面体。请说明多面体的程度。

$\begin{aligned} 16 x^{2} y^{3} - 3 x y^{5} - 2 x^{3} y^{2} + 2 x y - 7 x^{2} y^{3} + 2 x^{3} y^{2} \end{aligned}$
::16x2y3 - 3xy5 - 2x3y2+2xy - 7x2y3+2x3y2

Start by simplifying by combining like terms:
::从简化开始,将类似术语合并:

$\begin{aligned} 16 x^{2} y^{3} - 3 x y^{5} - 2 x^{3} y^{2} + 2 x y - 7 x^{2} y^{3} + 2 x^{3} y^{2} \end{aligned}$
::16x2y3 - 3xy5 - 2x3y2+2xy - 7x2y3+2x3y2

is equal to
::等于

$\begin{aligned} (16 x^{2} y^{3} - 7 x^{2} y^{3}) - 3 x y^{5} + (- 2 x^{3} y^{2} + 2 x^{3} y^{2}) + 2 x y \end{aligned}$
:16x2y3-7x2y3)-3x5+(-2x3y2+2x3y2)+2xy2)

which simplifies to
::简化到

$\begin{aligned} 9 x^{2} y^{3} - 3 x y^{5} + 2 x y . \end{aligned}$
::9x2y3 - 3xy5+2xy.

In order to rewrite in standard form, we need to determine the degree of each term. The first term has a degree of $\begin{aligned} 2 + 3 = 5 \end{aligned}$ , the second term has a degree of $\begin{aligned} 1 + 5 = 6 \end{aligned}$ , and the last term has a degree of $\begin{aligned} 1 + 1 = 2 \end{aligned}$ . We will rewrite the terms in order from largest degree to smallest degree:
::为了以标准格式重写,我们需要确定每个任期的级别。第一个任期为2+3=5,第二个任期为1+5=6,最后一个任期为1+1=2。我们将从最大到最小的顺序重写术语:

$\begin{aligned} - 3 x y^{5} + 9 x^{2} y^{3} + 2 x y \end{aligned}$
::- 3xy5+9x2y3+2xy

The degree of a polynomial is the largest degree of all the terms. In this case that is 6.
::多元度是所有术语中的最大程度,在此情况下是6。

Review
::回顾

Indicate whether each expression is a polynomial.
::指示每个表达式是否是一个多面性表达式。

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

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

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

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

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

$\begin{aligned} {(x^{\frac{3}{2}})}^{2} \end{aligned}$
:x32)2

Express each polynomial in standard form. Give the degree of each polynomial.
::以标准格式表达每个多面体。给每个多面体的程度。

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

$\begin{aligned} 8 - 4 x + 3 x^{3} \end{aligned}$
::8-4x+3x3 8-4x+3x3

$\begin{aligned} - 5 + 2 x - 5 x^{2} + 8 x^{3} \end{aligned}$
::- 5+2x-5x2+8x3

$\begin{aligned} x^{2} - 9 x^{4} + 12 \end{aligned}$
::x2- 9x4+12 x2- 9x4+12

$\begin{aligned} 5 x + 2 x^{2} - 3 x \end{aligned}$
::5x+2x2-2-3x

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

General

标准表格中的多面体

Polynomials in Standard Form ::标准表格中的多面体

Identifying Constants and the Degree of a Polynomial ::识别常数和多元度

Identifying Polynomials ::识别多面体

Writing Polynomials in Standard Form ::在标准表格中写入多面体

Simplifying Polynomials ::简化多面体

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Polynomials in Standard Form
::标准表格中的多面体

Identifying Constants and the Degree of a Polynomial
::识别常数和多元度

Identifying Polynomials
::识别多面体

Writing Polynomials in Standard Form
::在标准表格中写入多面体

Simplifying Polynomials
::简化多面体

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)