课程： 9.6 一夫多妻制的单一因素

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一夫多妻制的单一因素
Monomial Factors of Polynomials
::一夫多妻制的单一因素

In the last few sections, we learned how to multiply by using the . All the terms in one polynomial had to be multiplied by all the terms in the other polynomial. In this section, you’ll start learning how to do this process in reverse. The reverse of distribution is called factoring .
::在最后几节中,我们学会了如何通过使用来乘法。在一个多边协议中的所有术语都必须乘以另一个多边协议中的所有术语。在本节中,你将开始学习如何逆向地完成这一过程。分配的反向是所谓的乘数。

The total area of the figure above can be found in two ways.
::上图的总面积有两种方式。

We could find the areas of all the small rectangles and add them: $\begin{aligned} a b + a c + a d + a e + 2 a \end{aligned}$ .
::我们可以找到所有小矩形的区域并添加:ab+ac+ad+ae+2a。

Or, we could find the area of the big rectangle all at once. Its width is $\begin{aligned} a \end{aligned}$ and its length is $\begin{aligned} b + c + d + e + 2 \end{aligned}$ , so its area is $\begin{aligned} a (b + c + d + e + 2) \end{aligned}$ .
::或者,我们可以同时找到大矩形的区域。其宽度为a, 长度为 b+c+d+e+2, 因此其区域为a( b+c+d+e+2) 。

Since the area of the rectangle is the same no matter what method we use, those two expressions must be equal.
::由于矩形区域是相同的,无论我们使用何种方法,这两个表达方式必须相同。

$\begin{aligned} a b + a c + a d + a e + 2 a = a (b + c + d + e + 2) \end{aligned}$

::a b+ac+ad+ae+2a=a(b+c+d+e+2)

To turn the right-hand side of this equation into the left-hand side, we would use the distributive property. To turn the left-hand side into the right-hand side, we would need to factor it. Since polynomials can be multiplied just like numbers, they can also be factored just like numbers—and we’ll see later how this can help us solve problems.
::将这个等式的右侧转换为左侧,我们将使用分配属性。要将左侧转换为右侧,我们需要将它考虑在内。由于多面体可以像数字一样乘以多个数字,它们也可以像数字一样计算 — — 我们稍后会看到这如何帮助我们解决问题。

Find the Greatest Common Monomial Factor
::寻找最大的共同单一因素

You will be learning several factoring methods in the next few sections. In most cases, factoring takes several steps to complete because we want to factor completely . That means that we factor until we can’t factor any more.
::在接下来的几节中,你将学习几种保理方法。在多数情况下,保理方法需要几个步骤才能完成,因为我们想完全考虑。这意味着在我们无法再考虑之前我们要考虑。

Let’s start with the simplest step: finding the greatest monomial factor. When we want to factor, we always look for common monomials first. Consider the following polynomial, written in expanded form :
::让我们从最简单的步骤开始:找到最大的单一因子。当我们想考虑时,我们总是首先寻找共同的单一因子。考虑以下以扩展形式写成的多元因子:

$\begin{aligned} a x + b x + c x + d x \end{aligned}$

::ax+bx+cx+dx

A common factor is any factor that appears in all terms of the polynomial; it can be a number, a variable or a combination of numbers and variables. Notice that in our example, the factor $\begin{aligned} x \end{aligned}$ appears in all terms, so it is a common factor.
::共同系数是指从多数值的所有角度出现的任何系数;它可以是数字、变量或数字和变量的组合。请注意,在我们的例子中,系数 x 在所有术语中都出现,因此它是一个共同系数。

To factor out the $\begin{aligned} x \end{aligned}$ , we write it outside a set of " data-term="Parentheses" role="term" tabindex="0"> parentheses . Inside the parentheses, we write what’s left when we divide each term by $\begin{aligned} x \end{aligned}$ :
::以乘以 x, 我们把它写在一组括号之外。在括号内, 我们写下当我们将每个术语除以 x 时留下的东西 :

$\begin{aligned} x (a + b + c + d) \end{aligned}$

:a+b+c+d)

Let’s look at more examples.
::让我们再看看更多的例子。

Factoring
::保理

Factor:
::因素 :

a) $\begin{aligned} 2 x + 8 \end{aligned}$
::a) 2x+8

We see that the factor 2 divides evenly into both terms: $\begin{aligned} 2 x + 8 = 2 (x) + 2 (4) \end{aligned}$
::我们认为,系数2在两个术语中均匀地分为2x+8=2(x)+2(4)。

We factor out the 2 by writing it in front of a parenthesis: $\begin{aligned} 2 () \end{aligned}$
::我们通过在括号前写出2 来算出2 :2

Inside the parenthesis we write what is left of each term when we divide by 2: $\begin{aligned} 2 (x + 4) \end{aligned}$
::在括号内,当我们除以 2: 2 (x+4) 时, 我们写下每个术语的剩余部分。

b) $\begin{aligned} 15 x - 25 \end{aligned}$
:b) 15x-25

We see that the factor of 5 divides evenly into all terms: $\begin{aligned} 15 x - 25 = 5 (3 x) - 5 (5) \end{aligned}$
::我们看到,5个系数在所有术语中均匀划分:15x-25=5(3x)-5(5)。

Factor out the 5 to get: $\begin{aligned} 5 (3 x - 5) \end{aligned}$
::乘以5 获得: 5(3x-5)

c) $\begin{aligned} 3 a + 9 b + 6 \end{aligned}$
:c) 3a+9b+6

We see that the factor of 3 divides evenly into all terms: $\begin{aligned} 3 a + 9 b + 6 = 3 (a) + 3 (3 b) + 3 (2) \end{aligned}$
::我们发现,3个系数在所有术语中均分:3a+9b+6=3(a)+3(3b)+3(2)。

Factor 3 to get: $\begin{aligned} 3 (a + 3 b + 2) \end{aligned}$
::因素3 获得:3(a+3b+2)

Finding the Greatest Common Factor
::寻找最伟大的共同因素

a) $\begin{aligned} a^{3} - 3 a^{2} + 4 a \end{aligned}$
::a) a3-3a2+4a

Notice that the factor $\begin{aligned} a \end{aligned}$ appears in all terms of $\begin{aligned} a^{3} - 3 a^{2} + 4 a \end{aligned}$ , but each term has $\begin{aligned} a \end{aligned}$ raised to a different power. The greatest common factor of all the terms is simply $\begin{aligned} a \end{aligned}$ .
::注意一个因素在a3-3a2+4a的所有术语中都出现,但每个术语都具有不同的权力,所有术语中最大的共同因素只是一个。

So first we rewrite $\begin{aligned} a^{3} - 3 a^{2} + 4 a \end{aligned}$ as $\begin{aligned} a (a^{2}) + a (- 3 a) + a (4) \end{aligned}$ .
::因此,我们首先将a3-3a2+4a改写为a2+a(3a)+a(4)。

Then we factor out the $\begin{aligned} a \end{aligned}$ to get $\begin{aligned} a (a^{2} - 3 a + 4) . \end{aligned}$
::然后我们算出一个得到一个(a2-3a+4)的系数。

b) $\begin{aligned} 12 a^{4} - 5 a^{3} + 7 a^{2} \end{aligned}$
:b) 12a4-5a3+7a2

The factor $\begin{aligned} a \end{aligned}$ appears in all the terms, and it’s always raised to at least the second power. So the greatest common factor of all the terms is $\begin{aligned} a^{2} \end{aligned}$ .
::一个因素在所有术语中都出现,它总是被提升到至少第二权力。因此,所有术语中最大的共同因素是A2。

We rewrite the expression $\begin{aligned} 12 a^{4} - 5 a^{3} + 7 a^{2} \end{aligned}$ as $\begin{aligned} (12 a^{2} \cdot a^{2}) - (5 a \cdot a^{2}) + (7 \cdot a^{2}) \end{aligned}$
::我们改写12a4-5a3+7a2(12a2a2)-(5aa2)+(7a2)

Factor out the $\begin{aligned} a^{2} \end{aligned}$ to get $\begin{aligned} a^{2} (12 a^{2} - 5 a + 7) \end{aligned}$ .
::乘以 A2 以获得 a2( 12a2- 5a+7) 。

Complete Factoring
::完整保理

Factor completely:
::完全因数 :

a) $\begin{aligned} 3 a x + 9 a \end{aligned}$
::a) 3ax+9a

Both terms have a common factor of 3, but they also have a common factor of $\begin{aligned} a \end{aligned}$ . It’s simplest to factor these both out at once, which gives us $\begin{aligned} 3 a (x + 3) \end{aligned}$ .
::这两个词都有3个共同系数,但它们也有1个共同系数。最简单的因素是同时将两者都排除在外,这给了我们3a(x)+3。

b) $\begin{aligned} x^{3} y + x y \end{aligned}$
::b) x3y+xy

Both $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ are common factors. When we factor them both out at once, we get $\begin{aligned} x y (x^{2} + 1) \end{aligned}$ .
::x 和 y 均为共同因素。当我们同时将两者都考虑在内时, 我们就会得到xy( x2+1 ) 。

c) $\begin{aligned} 5 x^{3} y - 15 x^{2} y^{2} + 25 x y^{3} \end{aligned}$
:c) 5x3y-15x2y2+25xy3

The common factors are 5, $\begin{aligned} x \end{aligned}$ , and $\begin{aligned} y \end{aligned}$ . Factoring out $\begin{aligned} 5 x y \end{aligned}$ gives us $\begin{aligned} 5 x y (x^{2} - 3 x y + 5 x y^{2}) \end{aligned}$ .
::常见系数是 5 x 和 Y. 乘以 5xy 给我们5xy(x2-3xy+5xy2)。

Example
::示例示例示例示例

Example 1
::例1

Find the greatest common factor.
::找到最大的共同因素。

$\begin{aligned} 16 x^{2} y^{3} z^{2} + 4 x^{3} y z + 8 x^{2} y^{4} z^{5} \end{aligned}$
::16x2y3z2+4x3yz+8x2y4z5

First, look at the coefficients to see if they share any common factors. They do: 4.
::首先,看看系数,看它们是否具有任何共同因素,它们有:4。

Next, look for the lowest power of each variable, because that is the most you can factor out. The lowest power of $\begin{aligned} x \end{aligned}$ is $\begin{aligned} x^{2} \end{aligned}$ . The lowest powers of $\begin{aligned} y \end{aligned}$ and $\begin{aligned} z \end{aligned}$ are to the first power.
::下一步, 查找每个变量的最低功率, 因为这是您能考虑到的最大功率。 x 的最低功率是 x2 。 y 和 z 的最低功率是第一功率。

This means we can factor out $\begin{aligned} 4 x^{2} y z \end{aligned}$ . Now, we have to determine what is left in each term after we factor out $\begin{aligned} 4 x^{2} y z \end{aligned}$ :
::这意味着我们可以把4x2yz算出来。现在,我们必须在把4x2yz算出来之后,确定每个术语的剩余部分:

$\begin{aligned} 16 x^{2} y^{3} z^{2} + 4 x^{3} y z + 8 x^{2} y^{4} z^{5} = 4 x^{2} y z (4 y^{2} z + x + 2 y^{3} z^{4}) \end{aligned}$
::16x2y3z2+4x3z+8x2y4z5=4x2yz(4y2z+x+2y3z4)

Review
::回顾

Factor out the greatest common factor in the following polynomials.
::在以下多面体中计出最大的共同因素。

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

$\begin{aligned} 3 x^{3} - 21 x \end{aligned}$
::3x3 - 21x

$\begin{aligned} 5 x^{6} + 15 x^{4} \end{aligned}$
::5x6+15x4

$\begin{aligned} 4 x^{3} + 10 x^{2} - 2 x \end{aligned}$
::4x3+10x2-2x

$\begin{aligned} - 10 x^{6} + 12 x^{5} - 4 x^{4} \end{aligned}$
::- 10x6+12x5-4x4

$\begin{aligned} 12 x y + 24 x y^{2} + 36 x y^{3} \end{aligned}$
::12xy+24xy2+36xy3

$\begin{aligned} 5 a^{3} - 7 a \end{aligned}$
::5a3-7a 5a3-7a

$\begin{aligned} 3 y + 6 z \end{aligned}$
::3y+6z 3y+6z

$\begin{aligned} 10 a^{3} - 4 a b \end{aligned}$
::10a3-4ab

$\begin{aligned} 45 y^{12} + 30 y^{10} \end{aligned}$
::45y12+30y10

$\begin{aligned} 16 x y^{2} z + 4 x^{3} y \end{aligned}$
::16xy2z+4x3y

$\begin{aligned} 2 a - 4 a^{2} + 6 \end{aligned}$
::2a-4a2+6

$\begin{aligned} 5 x y^{2} - 10 x y + 5 y^{2} \end{aligned}$
::5xy2 - 10xy+5y2

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。

章节大纲

常规

一夫多妻制的单一因素

Monomial Factors of Polynomials ::一夫多妻制的单一因素

Find the Greatest Common Monomial Factor ::寻找最大的共同单一因素

Factoring ::保理

Finding the Greatest Common Factor ::寻找最伟大的共同因素

Complete Factoring ::完整保理

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Monomial Factors of Polynomials
::一夫多妻制的单一因素

Find the Greatest Common Monomial Factor
::寻找最大的共同单一因素

Factoring
::保理

Finding the Greatest Common Factor
::寻找最伟大的共同因素

Complete Factoring
::完整保理

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)