Section: 考虑最大共同因素 | 7.2 考虑最大共同因素

Section outline

The revenue of a company is the amount of money the company receives for selling a product. Say the revenue of a company that sells video games , in thousands of dollars, is $\begin{aligned} R (x) = - x^{2} + 400 x \end{aligned}$ , where x is the numbers of products sold. Being able to factor this function can help us determine when the company has $0 in revenue, as we will see in the next chapter. We start to factor expressions in this section.
::公司的收入是公司出售产品所得的金额。说出售视频游戏的公司以千美元计的收入是R(x)x2+400x, 其中x是销售产品的数量。能够将这一功能考虑在内,可以帮助我们确定公司何时有零美元收入,我们将在下一章中看到这一点。我们开始考虑本节中的表达方式。

Factoring Algebraic Expressions
::乘数代数表达式

When we factor numbers, we can write a number as a product of factors—for example, $\begin{aligned} 36 = 3 \cdot 12 \end{aligned}$ —or we can find the prime factorization of the number. The prime factorization of $\begin{aligned} 36 = 2 \cdot 2 \cdot 3 \cdot 3 = 2^{2} \cdot 3^{2} \end{aligned}$ .
::当我们使用系数编号时,我们可以以因数(例如,36=312)的产物来写一个数字,或者我们能找到数字的基因化。36=223=2232的基因化。

We have an analogous idea for factoring algebraic expressions.
::我们对代数表达式的乘法有一个类似的想法。

Factoring Completely
::全额保理

An algebraic expression is factored completely when it is a product of prime factors.
::当代数表达式是主要因素的产物时,就将其完全考虑在内。

If an algebraic expression cannot be written as a product of two other algebraic expressions (excluding 1 and -1), then the expression is prime .
::如果代数表达式不能作为另外两个代数表达式(不包括1和-1)的产物写入,则该表达式为主表达式。

Throughout this chapter, our goal is to factor completely.
::在整个本章中,我们的目标是完全考虑。

Identifying a Greatest Common Factor (GCF)
::确定最大共同因素(GCF)

The greatest common factor of an algebraic expression is the "greatest" factor that can be divided into each of the terms with no remainder. We will use the same techniques to find the GCF of a number, and introduce techniques for finding the GCF of terms with variables.
::代数表达的最大共同因素是“最大”因素,可以分为每个术语,无所余。我们将使用同样的技术来寻找数字的绿色气候基金,并引入技术来寻找与变量条件的绿色气候基金。

Example 1
::例1

Find the GCF of the following terms: $\begin{aligned} 18 x^{3} y, 24 x^{2} y^{3}, 36 x^{4} y^{2} \end{aligned}$ .
::绿色气候基金有以下条件:18x3y,24x2y3,36x4y2。

Solution: To find the GCF of terms with variables, we consider the numbers and each variable separately. First, the prime factorization of each of the numbers is
::解决办法:为找到以变量为条件的绿色气候基金,我们分别考虑数字和每个变量。

$\begin{aligned} 18 & = 2 \cdot 3^{2} \\ 24 & = 2^{3} \cdot 3 \\ 36 & = 2^{2} \cdot 3^{2} \end{aligned}$

The GCF is the product of the factors that appear in each number. At least one 2 and one 3 are factors of each number, so the GCF is $\begin{aligned} 2 \cdot 3 = 6 \end{aligned}$ .
::全球合作框架是每个数字中出现的各种因素的产物,每个数字中至少有一个2个和1个3个因素,因此全球合作框架为2 3=6。

Now we turn to the x 's. The 1st term has 3, the 2nd has 2, and the 3rd has 4. Each term contains at least two x 's, so the GCF is $\begin{aligned} x^{2} . \end{aligned}$
::现在我们转向 x。第一个学期是 3, 第二个学期是 2, 第三个学期是 4, 每个学期至少包含 2 x , 所以全球合作框架是 x 2 。

Lastly, we have the y 's. The 1st term has 1, the 2nd has 3, and the 3rd has 2. Each term contains at least one y , so the GCF is y.
::最后,我们有y。第一个学期是1,第二个学期是3,第三个学期是2,每个学期至少包括一个y,所以全球合作框架就是y。

Together, the GCF is the product: $\begin{aligned} 6 \cdot x^{2} \cdot y = 6 x^{2} y \end{aligned}$ .
::全球合作框架的产物是:6x2y=6x2y。

by Mathispower4u demonstrates how to determine the greatest common factor of two monomials.
::Mathispower4u 展示了如何确定两个单子体的最大共同系数。

Factoring Out the GCF
::利用绿色气候基金

Factoring out a GCF means writing our expression as we do the . Recall (from Chapter 6, Section 4), the distributive property states that $\begin{aligned} a (b + c) = a \cdot b + a \cdot c \end{aligned}$ . For our purposes now, the a in the distributive property plays the role of the GCF. We want to place it on the outside of the " data-term="Parentheses" role="term" tabindex="0"> parentheses to multiply by it. Inside the parentheses, we will divide each term by a to get b and c .
::将绿色气候基金纳入考虑范围,意味着我们像我们一样写下表达方式。回顾(第6章第4节),分配性财产指出,a(b+c)=ab+ac。为了我们现在的目的,分配性财产中的a发挥绿色气候基金的作用。我们希望将其置于括号外,以乘以它。在括号内,我们将每个术语除以a,以获得b和c。

$\begin{aligned} a \cdot b + a \cdot c = a (\frac{a b}{a} + \frac{a c}{a}) = a (b + c) \end{aligned}$

::a-b+a-c=a(aba+aca)=a(b+c)

By multiplying and dividing by the same term, we do not change the value of the expression, but just change how it looks.
::通过乘法和除法,我们不会改变表达式的价值,而只是改变其外观。

Example 2
::例2

The revenue of a company that sells video games , in thousands of dollars, is $\begin{aligned} R (x) = - x^{2} + 400 x \end{aligned}$ , where x is the numbers of products sold. Factor the revenue function.
::销售电子游戏的公司以千美元计的收入是R(x)=-x2+400x,其中x是销售产品的数量。

Solution: We need to factor $\begin{aligned} - x^{2} + 400 x \end{aligned}$ . Neither of the terms has a common factor that is a number, but it is often convenient to factor out -1. The 1st term has two x 's, and the 2nd has one. We can factor out one x . Together, we will factor out $\begin{aligned} - 1 \cdot x = - x \end{aligned}$ .
::解答 : 我们需要以乘数 -x2+400x 。这两个词没有一个共同的乘数, 但通常很容易以乘数计数 -1. 第一个词有两个乘数, 第二个词有一个。我们可以以乘数一个乘数 x. 我们一起以乘数 -1x=-x 。

$\begin{aligned} - x^{2} + 400 x = - x (\frac{- x^{2}}{- x} + \frac{400 x}{- x}) = - x (x - 400) \end{aligned}$

::-x2+400x=-x(-x2-x+400x-x)=-x(x-400)

This factored form can help us determine when the revenue will be 0. This is explained in the Explore More problems below in this section.
::这种因数形式可以帮助我们确定何时收入为0。这一点在下文本节《探讨更多问题》中作了解释。

Example 3
::例3

Factor $\begin{aligned} 16 z^{5} - 8 z^{3} + 12 z \end{aligned}$ .
::系数 16z5-8z3+12z。

Solution: We consider the GCF of the numbers and the z 's separately. The prime factorization of the numbers is
::解决方案:我们分别考虑数字和z的绿色气候基金。

$\begin{aligned} 16 & = 2^{4} \\ 8 & = 2^{3} \\ 12 & = 2^{2} \cdot 3 \end{aligned}$

There are at least two 2's in each number. There is also at least one z in each term. Together, the GCF is $\begin{aligned} 2^{2} \cdot z = 4 z \end{aligned}$ . Factoring that out of the polynomial above, we have
::每个数字中至少有两个 2 。每个术语中至少有一个 z。 GFO 总共是 22z=4z。从上面的多元数字中,我们考虑到这一点,

$\begin{aligned} 16 z^{5} - 8 z^{3} + 12 z = 4 z (\frac{16 z^{5}}{4 z} - \frac{8 z^{3}}{4 z} + \frac{12 z}{4 z}) = 4 z (4 z^{4} - 2 z^{2} + 3) . \end{aligned}$

::16z5-8z3+12z=4z(16z54z-8z34z+12z4z)=4z(4z4-2z2+3)。

by Mathispower4u demonstrates how to factor the greatest common factor out of a trinomial .
::Mathispower4u 展示了如何在三重力中将最大共同因素乘以。

Example 4
::例4

Factor $\begin{aligned} 3 a (4 a + b) + 5 b (4 a + b) \end{aligned}$ .
::系数3a(4a+b)+5b(4a+b)。

Solution: GCFs are not limited to one-term expressions. They can have multiple terms, like the binomial $\begin{aligned} 4 a + b \end{aligned}$ . Each of the terms above contains a factor of $\begin{aligned} 4 a + b \end{aligned}$ , so we can factor it out.
::解决方案:全球合作框架并不限于一期表达式。它们可以有多个术语, 如二元制 4a+b。以上每个术语包含一个 4a+b 的系数, 所以我们可以将其考虑在内。

$\begin{aligned} 3 a (4 a + b) + 5 b (4 a + b) = (4 a + b) (\frac{3 a (4 a + b)}{4 a + b} + \frac{5 b (4 a + b)}{4 a + b}) = (4 a + b) (3 a + 5 b) \end{aligned}$

::3a(4a+b)+5b(4a+b)=(4a+b)=(4a+b)=(4a+b)3a(4a+b)4a+b+5b(4a+b)4a+b)=(4a+b)(3a+5b)

Example 5
::例5

Factor $\begin{aligned} 2 x (x - 3) - 7 (3 - x) \end{aligned}$ .
::倍数 2x(x-3)- 7(3-x) 。

Solution: Sometimes, two expressions look similar, but are different due to a factor of -1. If you multiply $\begin{aligned} x - 3 \end{aligned}$ by -1, you get $\begin{aligned} - x + 3 = 3 - x \end{aligned}$ , which is the binomial in the 2nd term. If we factor a -1 out of the 2nd binomial, we get
::解决方案 : 有时, 两个表达式看起来相似, 但因 -1 系数而不同。如果您乘 x-3 乘 x- 3 乘乘 -1, 你就会得到 - x+3=3 -x, 这是第二个学期的二进制。如果我们将二二进制中的a-1乘以

$\begin{aligned} 2 x (x - 3) - 7 (3 - x) & = 2 x (x - 3) - 7 \cdot - 1 (- 3 + x) \\ = 2 x (x - 3) + 7 (x - 3) \\ = (x - 3) (2 x + 7) \end{aligned}$

::2x(x-3)-7(3-x)=2x(x-3)-7(3+x)=2x(3+x)=2x(x-3)+7(x-3)=(x-3)(2x+7)

This video by CK-12 demonstrates how to factor expressions by rearranging opposites and removing common binomials.
::CK-12的这段影片展示了如何通过重新排列对面面和去除常见二元论来进行要素表达。



Feature: Jump Height
::特点: 跳高

by Denise Huey
::丹妮丝·胡伊(Denise Huey)

Vertical jump height is discussed quite frequently among basketball players and fans. Vertical jump height refers to the distance between the highest point a person can reach after a big jump and the standing reach height.
::篮球运动员和球迷经常讨论垂直跳高。垂直跳高是指一个人在跳跃后能够达到的最高点与站立跳高之间的距离。

David Noel, from the NBA Draft of 2006, had a vertical jump height of 34 inches. The quadratic equation that best describes his vertical jump height over time is $\begin{aligned} h = 162 t - 192 t^{2} \end{aligned}$ , with $\begin{aligned} h \end{aligned}$ representing the vertical jump height, and $\begin{aligned} t \end{aligned}$ representing time in seconds. We can use this equation to find how long it took for him to land on the ground.
::大卫·诺埃尔(David Noel),来自2006年《美国国家律师协会草案》,垂直跳跃高度为34英寸。最能描述其纵向跳跃高度的二次方程式是:H=162t-192t2, 代表垂直跳跃高度, t代表时间的秒数。我们可以用这个方程式来寻找他降落在地面的时间。

Applying our understanding of factoring, we can factor out $\begin{aligned} 6 t \end{aligned}$ from both terms on the right side of the equation. Once we do that, we have $\begin{aligned} h = 6 t (27 - 32 t) \end{aligned}$ . We can use the zero p roduct property to set each factor equal to zero, and solve for $\begin{aligned} t \end{aligned}$ when $\begin{aligned} h = 0 \end{aligned}$ .
::应用我们对保理因素的理解, 我们可以从公式右侧的两个条件中计出 6t。一旦我们这样做, 我们就会有 h= 6t( 27- 32t) 。我们可以使用零产品属性来设定每个系数为零, 并在 h=0 时解决 t 。

$\begin{aligned} 0 = 6 t (27 - 32 t) \end{aligned}$ We factored $\begin{aligned} 6 t \end{aligned}$ from both terms because $\begin{aligned} 6 t \end{aligned}$ is the greatest common factor of both terms.
::0=6t(27-32t) 我们从两个术语中都计算了6t,因为6t是两个术语中最大的共同系数。

Using the z ero p roduct property, we can set each factor equal to zero and solve for $\begin{aligned} t \end{aligned}$ : $\begin{aligned} 6 t = 0 \end{aligned}$ and $\begin{aligned} 27 - 32 t = 0 \end{aligned}$ and $\begin{aligned} t = 0 \end{aligned}$ and $\begin{aligned} t = \frac{27}{32} = 0.84375 \end{aligned}$ , rounded to 0.84.
::使用零产品属性,我们可以将每个系数设定为零,解决 t:6t=0和27-32t=0和t=0和t=0和t=2732=0.84375,四舍五入为0.84。

What do these numbers mean? They mean that at 0 and 0.84 seconds, Noel’s vertical jump height is 0 feet. Why are there two numbers? The two numbers represent the time that Noel started at 0 seconds, and the time it took him to reach the ground after jumping.
::这些数字意味着什么?它们意味着在0和0.84秒时,诺埃尔的垂直跳跃高度为0英尺。为什么有两个数字?这两个数字代表诺埃尔在0秒时开始跳动的时间,以及他跳跃后到达地面的时间。

by Ryan Van Dusen shows Noel winning the 2006 Slam Dunk competition .
::Ryan Van Dusen展示诺埃尔在2006年的Slam Dunk比赛中获胜。

Summary
::摘要

To recognize a greatest common factor, find a greatest common factor for the numbers in the expression and then consider each variable or expression separately. If the variable or expression appears in all of the terms, factor out the smallest power that appears.
::要识别一个最大的共同因素, 找到表达式中数字的最大共同因素, 然后分别考虑每个变量或表达式。如果变量或表达式在所有术语中都出现, 请将显示的最小功率考虑在内。

To factor out a GCF, write the GCF outside the parentheses and divide each one of the terms by the GCF in the parentheses.
::将绿色气候基金考虑在内,在括号外写绿色气候基金,在括号内按绿色气候基金划分每个条件。

Review
::回顾

Factor the common factor from the following polynomials:
::乘以下列多数值得出的共同系数:

1. $\begin{aligned} 36 a^{2} + 9 a^{3} - 6 a^{7} \end{aligned}$
::1. 36a2+9a3-6a7

2. $\begin{aligned} y x^{3} y^{2} + 12 x + 16 y \end{aligned}$
::2. yx3y2+12x+16y

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

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

5. $\begin{aligned} 4 x^{3} + 10 x^{2} - 2 x \end{aligned}$
::5. 4x3+10x2-2x

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

7. $\begin{aligned} 12 x y + 24 x y^{2} + 36 x y^{3} \end{aligned}$
::7. 12xy+24xy2+36xy3

8. $\begin{aligned} 5 a^{3} - 7 a \end{aligned}$
::8. 5a3-7a

9. $\begin{aligned} 45 y^{12} + 30 y^{10} \end{aligned}$
::9. 45y12+30y10

10. $\begin{aligned} 16 x y^{2} z + 4 x^{3} y \end{aligned}$
::10. 16xy2z+4x3y

11. $\begin{aligned} 2 x (x + 3) + 4 (x + 3) \end{aligned}$
::11. 2x(x+3)+4(x+3)

12. $\begin{aligned} 4 y^{2} (3 y - 1) - 5 (3 y - 1) \end{aligned}$
::12. 4y2(3y-1-1)-5(3y-1)

13. $\begin{aligned} 7 a (8 b - 3) - 11 (3 - 8 b) \end{aligned}$
::13. 7a(8b-3)-11(3-8b)

14. $\begin{aligned} 3 z (z - 5) + 7 (5 - z) \end{aligned}$
::14. 3z(z-5)+7(5-z)

15. $\begin{aligned} 4 a (a + b + c) - 3 b (a + b + c) - 2 c (a + b + c) \end{aligned}$
::15. 4a(a+b+c)-3b(a+b+c)-2c(a+b+c)-2c(a+b+c)

Explore More
::探索更多

1. The surface area of a cylinder is $\begin{aligned} 2 π r^{2} + 2 π r h \end{aligned}$ , where r is the radius of the cylinder and h is the height. Write the surface area formula in factored form.
::1. 圆柱体的表面面积为2r2+2rh,其中r为圆柱体半径,h为高度。用乘数形式写上表面积公式。

2. The supply of a product is how much product a business can supply based on the price, p . If the supply function is $\begin{aligned} S (p) = 10 p + 5 p^{2}, \end{aligned}$ write the function in factored form.
::2. 产品的供应是企业能够根据价格提供多少产品,p。如果供应功能是S(p)=10p+5p2, 则按系数格式填写该功能。

3. The zero p roduct property states that if the result of multiplying factors together is 0, then at least one of the factors is 0. Algebraically, $\begin{aligned} a \cdot b = 0 \end{aligned}$ implies that $\begin{aligned} a = 0 \end{aligned}$ or $\begin{aligned} b = 0 \end{aligned}$ or both. Use the zero p roduct property to determine when the revenue from Example 2 is equal to 0.
::3. 零产品属性表示,如果乘数系数加在一起的结果为0,那么其中至少有一个系数为0。代数上, ab=0意味着a=0或b=0或两者兼而有之。使用零产品属性来确定例2的收入何时等于0。

Answers for Review and Explore More Problems
::回顾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:

Section outline

Factoring Algebraic Expressions ::乘数代数表达式

Factoring Completely ::全额保理

Identifying a Greatest Common Factor (GCF) ::确定最大共同因素(GCF)

Example 1 ::例1

Factoring Out the GCF ::利用绿色气候基金

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Feature: Jump Height ::特点: 跳高

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

Factoring Algebraic Expressions
::乘数代数表达式

Factoring Completely
::全额保理

Identifying a Greatest Common Factor (GCF)
::确定最大共同因素(GCF)

Example 1
::例1

Factoring Out the GCF
::利用绿色气候基金

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Feature: Jump Height
::特点: 跳高

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX