课程： 7.5 含铅系数等于1时的计数四方

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选择节当铅系数等于1时的四重力乘数

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当铅系数等于1时的四重力乘数
The area of the yard below can be expressed as $\begin{aligned} x^{2} - 3 x - 28 \end{aligned}$ . If we write this in factored form , we can determine the length and width of the yard. We discuss factoring quadratic expressions of this form below.
::下面院子的面积可以表示为 x2 - 3x - 28。如果我们以系数形式写下,我们可以确定院子的长度和宽度。我们下面讨论这种形式的弧度表示。

Factoring Quadratic Expressions When the Lead Coefficient Is 1
::当铅系数为 1 时计算二次曲线表达式

A quadratic expression is an expression of the form $\begin{aligned} a x^{2} + b x + c \end{aligned}$ , where a , b , and c are real numbers, or a 2nd- degree polynomial . We cannot factor all expressions of this form (in the real numbers). Those we can are highlighted in the box below.
::二次形表达式是表Ax2+bx+c的表达式,其中a、b和c为实际数字,或二度多圆形。我们不能将这种形式的所有表达式(以实际数字计)都考虑在内。我们可以在下面的框中突出显示这些表达式。

Factoring Quadratic Expressions When the Lead Coefficient Is 1
::当铅系数为 1 时计算二次曲线表达式

$\begin{aligned} x^{2} + (m + n) x + m n = (x + m) (x + n) \end{aligned}$

Our goal will be to find two integers whose product is c and that add up to b . If we cannot find them, the expression is prime. (We will not consider factoring more generally across or the complex numbers in this section.)
::我们的目标是找到两个整数,其产品为c,并且加起来为b。如果我们找不到, 表达式是首要的。 (我们将不考虑在本节中更笼统地考虑横跨或复杂的数字。 )

Example 1
::例1

Factor $\begin{aligned} x^{2} + 6 x + 8 \end{aligned}$ .
::系数 x2+6x+8

Solution: From the formula above, we see that we need two numbers that multiply to 8 whose sum is 6. Some people like to organize this search in an X- pattern , as shown below. We start by listing the factors of 8. Then we consider their sum.
::解决方案: 从上面的公式来看,我们看到我们需要两个数字, 乘以8, 其总和是6。有些人喜欢用 X 模式组织搜索, 如下文所示。我们首先列出 8 的因数。然后我们考虑他们的总和。

Note that 4 and 2 multiply out to 8 and add up to 6. Therefore , the factors of $\begin{aligned} x^{2} + 6 x + 8 \end{aligned}$ are $\begin{aligned} (x + 4) (x + 2) \end{aligned}$ . You can FOIL these binomials to check your answer.
::注意 4 和 2 乘以 8 和 6 。因此, x2+6x+8 的因数是 (x+4)(x+2) 。您可以通过 FIL 来查看您的答复。

Example 2
::例2

Factor $\begin{aligned} x^{2} + 12 x - 28 \end{aligned}$ .
::系数 x2+12x-28。

Solution: What are the factors of -28 that also add up to 12? Let's list the factors of 28:
::解决方案:什么是 - 28 的因素加起来也等于 12 ?让我们列出28 的因素:

$\begin{aligned} 1 \cdot 28 & 28 - 1 = 27 \\ 2 \cdot 14 & 14 - 2 = 12 \\ 4 \cdot 7 & 7 - 4 = 3 \end{aligned}$

Since the product is negative, the numbers will have opposite signs. When you add numbers with opposite signs, you subtract the absolute values and then take the sign of the larger number in absolute value . The $\begin{aligned} g r e e n \end{aligned}$ pair above is the one that works. We will keep 14 positive because the two numbers add up to positive 12.
::由于产品为负值, 数字将会有相反的信号。当你用相反的符号添加数字时, 您会减去绝对值, 然后以绝对值取较大数字的符号。上面的绿色对子是起作用的对子。我们将保持14个正数, 因为这两个数字加起来等于正12。

The factors are $\begin{aligned} x^{2} + 12 x - 28 = (x + 14) (x - 2) \end{aligned}$ .
::系数为 x2+12x-28=(x+14)(x-2)。

by CK-12 demonstrates how to factor quadratic expressions.
::通过 CK-12 演示如何乘以二次表达式。

Example 3
::例3

Factor $\begin{aligned} x^{2} - 9 x + 20 \end{aligned}$ .
::系数 x2-9x+20。

Solution: We need two numbers that multiply out to 20 and add up to -9. Let's focus on numbers that multiply out to 20.
::解答:我们需要两个数字乘以20,加起来乘以 -9. 让我们集中关注乘以20的数字

$\begin{aligned} 1 \cdot 20 & 1 + 20 = 21 \\ 2 \cdot 10 & 2 + 10 = 12 \\ 4 \cdot 5 & 4 + 5 = 9 \end{aligned}$

Since the two numbers multiply out to a positive number, they must have the same sign. When we add numbers with the same sign, we add their absolute values and keep the common sign. The 4 and 5 are the only two numbers that add up to 9. Since the 9 is negative, we have $\begin{aligned} - 4 + - 5 = - 9 \end{aligned}$ and $\begin{aligned} - 4 \cdot - 5 = 20 \end{aligned}$ . Thus, the factors are
::由于这两个数字乘以正数,它们必须有一个相同的符号。当我们用同一个符号加数字时,我们加上它们的绝对值并保留共同的符号。4和5是总共9个数字中唯一的两个数字。由于9个数字为负数,因此我们有 -4+5=9和 -45=20。因此,这些因素是-4+5=9和-4-5=20。

$\begin{aligned} x^{2} - 9 x + 20 = (x - 4) (x - 5) . \end{aligned}$

::x2-9x+20=(x-4)(x-5)-5。

Example 4
::例4

The area of the yard from the Introduction can be expressed as $\begin{aligned} x^{2} - 3 x + 28 \end{aligned}$ . Write this in factored form to find an expression for the length and for the width.
::引言后院子的面积可以以 x2-3x+28 表示。将此写成因数表达式, 以查找长度和宽度的表达式。

Solution: Recall that the area of a rectangle is $\begin{aligned} A = l w \end{aligned}$ , where l is the length and w is the width. To find the length and width, we can therefore factor the area $\begin{aligned} x^{2} - 3 x + 28 \end{aligned}$ .
::解析度: 提醒注意矩形的区域是 A=lw, 其中 I 是长度, w 是宽度。要找到长度和宽度, 我们可以因此将区域乘以 x2 - 3x+28 。

What are the factors of 28 that add up to 3? Testing the various possibilities
::28因素加到3是多少? 测试各种可能性

$\begin{aligned} 1 \cdot 28 & 1 + 28 = 29 \\ 2 \cdot 14 & 2 + 14 = 16 \\ 4 \cdot 7 & 7 - 4 = 3 \end{aligned}$

4 and 7 are the numbers, and since the sum is negative, we make 7 negative. We find that $\begin{aligned} - 7 \cdot 4 = - 28 \end{aligned}$ and $\begin{aligned} - 7 + 4 = - 3. \end{aligned}$ Therefore, $\begin{aligned} x^{2} - 3 x - 28 \end{aligned}$ factors to $\begin{aligned} (x - 2) (x - 14) \end{aligned}$ , and one of these factors is the rectangle's length while the other is its width.
::4和7是数字,由于总和为负数,我们得出7个负数,我们发现-74=28和-7+4=3。因此,X2-3x-28系数是(x-2)(x-14),其中一个因素是矩形长度,另一个是宽度。

This video by CK-12 demonstrates how to factor a trinomial when b and c are negative.
::这段影片由CK-12拍摄,

Example 5
::例5

Factor $\begin{aligned} 3 x^{2} + 21 x - 90 \end{aligned}$ .
::3x2+21x-90系数。

Solution: The 1st thing to look for when you are factoring is a GCF. Here, each one of the factors can be divided evenly by 3. We factor that out first: $\begin{aligned} 3 (x^{2} + 7 x - 30) \end{aligned}$ .
::解决方案:当您在计算系数时,首先需要寻找的是全球合作框架。在这里,每一个因素都可以平均地除以3。我们首先考虑的因素是:3(x2+7x-30)。

Now, let's list all the factors of 30 and their differences to see if the quadratic expression in " data-term="Parentheses" role="term" tabindex="0"> parentheses is factorable.
::现在,让我们列出所有30个因素及其差异, 看看括号中的四边表达式是否具有因数。

$\begin{aligned} 1 \cdot 30 & 30 - 1 = 29 \\ 2 \cdot 15 & 15 - 2 = 13 \\ 3 \cdot 10 & 10 - 3 = 7 \\ 5 \cdot 6 & 6 - 5 = 1 \end{aligned}$

From this, we see that the factors of -30 that add up to 7 are -3 and 10: $\begin{aligned} x^{2} + 7 x - 30 = (x - 3) (x + 10) . \end{aligned}$
::由此可见,30-30系数加7是-3和10:x2+7x-30=(x-3)(x+10)。

Example 6
::例6

Factor $\begin{aligned} x^{2} + x + 6 \end{aligned}$ .
::系数 x2+x+6

Solution: There are no factors of positive 6 that add up to positive 1. The factors of positive 6 include: 1 and 6, -1 and -6, 2 and 3, -2 and -3. None of these pairs of factors add to positive 1. Therefore, this is not a factorable trinomial.
::1. 积极6的因素包括:1和6、6-1和6、6、2和3、2和3。这些因素无一增加积极1。因此,这不是一个可考虑因素的三重因素。

by Mathispower4u demonstrates how to factor a polynomial when the leading coefficient is 1.
::通过 Mathispower4u 演示当主要系数为 1 时如何乘以多元系数。

Feature: Boiling Point
::特征:沸点

by Denise Huey
::丹妮丝·胡伊(Denise Huey)

At areas that are at or close to sea level, the boiling point of water is 212° F. The higher above sea level you are, the lower the boiling point is, since the air is thinner at higher altitudes. This becomes an issue for cooking at higher altitudes, especially at altitudes above 2,000 feet.
::在海平面上或接近海平面的地区,沸水点为212°F。海平面越高,沸水点越低,因为高空空气越薄,因此沸水点就越低,这成为高海拔地区烹饪的问题,特别是在2 000英尺以上的高度。

Scientists have used an approximate formula to find the boiling point of water at high altitudes. The formula is
::科学家使用一种近似公式在高空找到沸水点。

$\begin{aligned} D^{2} + 520 D = H . \end{aligned}$

::D2+520D=H.

$\begin{aligned} D \end{aligned}$ represents the degrees below 212 ° in which water will boil, and $\begin{aligned} H \end{aligned}$ represents the altitude above sea level.
::D表示水将沸腾的温度在212°以下,H表示海平面的海拔。

The city of Custer, South Dakota, sits at 5,300 feet above sea level. We can use this formula to find the point at which water will boil.
::南达科他州的卡斯特市位于海拔5 300英尺处,我们可以使用这个公式找到水沸腾的点。

$\begin{aligned} D^{2} + 520 D & = H \\ D^{2} + 520 D & = 5, 300 \\ D^{2} + 520 D - 5, 300 & = 0 \end{aligned}$

::D2+520D=HD2+520D=5,300D2+520D=5,300D-520D=0

We have set this equation equal to 0 to solve for $\begin{aligned} D \end{aligned}$ by factoring this quadratic. We know that the factors of 5,300 that have a difference of 520 are 530 and 10, so we will use these numbers in our factors.
::我们为D设定了等于0的等式,通过乘以这个二次方位来解答D。我们知道,5,300个差520个的系数是530个和10个,所以我们将用这些数字来计算我们的因素。

$\begin{aligned} D^{2} + 520 D - 5, 300 & = 0 \\ (D + 530) (D - 10) & = 0 \end{aligned}$

::D2+520D-5300=0(D+530)(D-10)=0

Now that we have factored our quadratic expression, we can set each factor equal to 0 and find the possible values of $\begin{aligned} D \end{aligned}$ .
::现在,我们已经将我们的二次表达式计算在内, 我们可以将每个系数设定为 0, 并找到D 的可能值。

$\begin{aligned} D + 530 = 0 & and D - 10 = 0 \\ D = - 530 & and D = 10 \end{aligned}$

::D+530=0和D-10=0D=530和D=10

Do both of these answers make sense for what we were trying to find? For finding the boiling points at high altitudes, the answer -530 does not make sense because $\begin{aligned} D \end{aligned}$ represents the degrees below 212 that water will boil: $\begin{aligned} 212 - (- 530) = 842 \end{aligned}$ , which is not below 212. On the other hand, $\begin{aligned} D = 10 \end{aligned}$ makes sense because this means that at Custer, water will boil 10 degrees below 212, or 212 – 10 or 202 degrees.
::这两种答案对我们试图找到的答案都有意义吗?为了在高空找到沸点,答案 - 530不合理,因为D代表着沸水的温度低于212度的温度:212-(-530)=842,这不低于212。另一方面,D=10意味着在库斯特,水的温度将低于212度,或212 - 10 或 202度。

One scientist created a shortcut and said that the boiling point will be 1 degree lower for every 500-foot increase. Does this work with Custer? Would this rule hold true for areas at different altitudes?
::一位科学家创造了一条捷径,并说沸点每增加500英尺就会降低1度。这与卡斯特有关吗?这一规则是否适用于不同高度的地区?

by Wayne Breslyn discusses the science behind pressure and temperature.
::Wayne Breslyn讨论压力和温度背后的科学。

Summary
::摘要

To factor quadratic expressions of the form $\begin{aligned} a x^{2} + b x + c \end{aligned}$ where $\begin{aligned} a = 1 \end{aligned}$ , you need to find a pair of numbers whose product is c and whose sum is b.
::在 a=1 的情况下,您需要找到产品为 c 且其总和为 b 的一对数字来表示窗体 ax2+bx+c 的二次形表达式。

Review
::回顾

Factor the expressions below . If it cannot be factored, write not factorable .
::以下的表达式因子。如果无法计算, 写不可计算。

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

2. $\begin{aligned} x^{2} + 2 x - 24 \end{aligned}$
::2. x2+2x-24

3. $\begin{aligned} x^{2} - 6 x \end{aligned}$
::3. x2-6x

4. $\begin{aligned} x^{2} + 6 x + 9 \end{aligned}$
::4. x2+6x+9

5. $\begin{aligned} x^{2} + 8 x - 10 \end{aligned}$
::5. x2+8x-10

6. $\begin{aligned} x^{2} - 11 x + 30 \end{aligned}$
::6. x2 - 11x+30

7. $\begin{aligned} x^{2} + 13 x - 30 \end{aligned}$
::7. x2+13x-30

8. $\begin{aligned} x^{2} + 11 x + 28 \end{aligned}$
::8. x2+11x+28

9. $\begin{aligned} x^{2} - 8 x + 12 \end{aligned}$
::9. x2-8x+12

10. $\begin{aligned} x^{2} - 7 x - 44 \end{aligned}$
::10. x2-7x-44

11. $\begin{aligned} x^{2} - 8 x - 20 \end{aligned}$
::11. x2-8x-20

12. $\begin{aligned} x^{2} + 4 x + 3 \end{aligned}$
::12. x2+4x+3

13. $\begin{aligned} x^{2} - 5 x + 36 \end{aligned}$
::13. x2 - 5x+36

14. $\begin{aligned} x^{2} - 5 x - 36 \end{aligned}$
::14. x2 - 5x-36

15. $\begin{aligned} x^{2} + x \end{aligned}$
::15. x2+x 15x2+x

Explore More
::探索更多

1. Why should you find the two numbers that multiply out to c 1st and not focus on the sum b ? What would happen if we did this the other way around and focused on the sum 1st?
::1. 为什么你要找到两个数字,这两个数字会乘以C1,而不是集中在A和B上? 如果我们这样做,然后把注意力放在A和B1上,会发生什么情况?

2. The techniques in this section can be used to factor perfect square trinomials and difference of two squares. Try to factor $\begin{aligned} x^{2} - 12 x + 36 \end{aligned}$ and $\begin{aligned} x^{2} - 64 \end{aligned}$ using the techniques from the previous section and the techniques from this section. Which method do you prefer? Why?
::2. 本节中的技术可用于计算完美的平方三角和两平方之差。尝试使用上一节中的技术和本节中的技术来计算乘数 x2-12x+36和 x2-64。您喜欢哪种方法?为什么?

by Mathispower4u provides examples of how to factor polynomials that require factoring out the GCF as the 1st step. Then other methods are used to completely factor the polynomial.
::由 Mathispower4u 提供示例,说明如何将需要将绿色气候基金作为第一级因素的多元性因素考虑在内。然后使用其他方法将多元性因素完全考虑在内。

3. One leg of a right triangle is 7 more than the other leg. The hypotenuse is 17. Find the dimensions of the triangle.
::3. 右三角的一条腿比另一条腿多7条,下限为17条,找出三角的尺寸。

by CK-12 below presents a similar real-world problem that is solved by factoring.
::下面的CK-12提出了一个类似的现实世界问题,通过保理解决。

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:

章节大纲

常规

当铅系数等于1时的四重力乘数

Factoring Quadratic Expressions When the Lead Coefficient Is 1 ::当铅系数为 1 时计算二次曲线表达式

Factoring Quadratic Expressions When the Lead Coefficient Is 1 ::当铅系数为 1 时计算二次曲线表达式

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Feature: Boiling Point ::特征:沸点

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 Quadratic Expressions When the Lead Coefficient Is 1
::当铅系数为 1 时计算二次曲线表达式

Factoring Quadratic Expressions When the Lead Coefficient Is 1
::当铅系数为 1 时计算二次曲线表达式

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Feature: Boiling Point
::特征:沸点

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