课程： 7.4 计算立方体的总和和差额

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计算立方体之和和之差
You are trying to find the right amount of insulation for a room that has the same length, width, and height. It is a cube-shaped room. The insulation fits between the outer wall of the room and the inner wall of the room. We can write an expression to model the amount of insulation, the volume , needed to cover the room. We can also factor this expression.
::您正在试图为一个长度、宽度和高度相同的房间找到合适的隔热层。这是一个立方体形的房间。隔热层适合房间外墙与房间内墙之间的隔热层。我们可以写一个表达式来模拟隔热层、体积、覆盖房间所需的隔热层数量。我们还可以将这个表达式考虑在内。

Sum and Difference of Two Cubes
::两个立方体的总和和和差额

Much like the difference of two squares that we looked at in the previous section, we have special factored forms for the sum and difference of two cubes.
::与我们在前一节中看到的两个方形的差别非常相似,我们对于两个立方体的总和和和差有特殊的因子表。

Sum and Difference of Two Cubes
::两个立方体的总和和和差额

$\begin{aligned} a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2}) \\ a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2}) \end{aligned}$

Unfortunately, you have to memorize the patterns. The video below may help with memorizing the forms. (If you would like to know why these factors work, see the Connection in this chapter.)
::不幸的是,您必须记住这些图案。下面的视频可能会有助于记住表格。 (如果您想知道这些因素为何有效,请参见本章的“连接”部分。 )

by Mathispower4u demonstrates how to factor the sum or difference of cubes.
::由 Mathispower4u 演示如何计算立方体的总和或差数。

Example 1
::例1

Factor $\begin{aligned} 8 x^{3} + 27 \end{aligned}$ .
::系数8x3+27。

Solution: First, determine if these are perfect cubes. A perfect cube is the result of raising an integer to the 3rd power; for example, $\begin{aligned} 1^{3} = 1, 2^{3} = 8, 3^{3} = 27, 4^{3} = 64, 5^{3} = 125, \end{aligned}$ and so on. Since 8 and 27 are perfect cubes, we can use a similar setup to the one in the previous section.
::解决方案 : 首先, 确定这些是否为完美的立方体。一个完美的立方体是将整数加到第三电源的结果; 例如, 13=1, 23=8, 33=27, 43=64, 53=125, 等等。由于 8 和 27 是完美的立方体, 我们可以使用与上一节相同的设置。

$\begin{aligned} a^{3} & = 8 x^{3} = (2 x)^{3} b^{3} = 27 = 3^{3} \\ a & = 2 x b = 3 \end{aligned}$

::a3=8x3=(2x)3b3=27=33a=2xb=3

Now, using the formula , we have:
::现在,使用公式,我们有:

$\begin{aligned} (a + b) (a^{2} - a b + b^{2}) & = (2 x + 3) ((2 x)^{2} - (2 x) (3) + 3^{2}) \\ = (2 x + 3) (4 x^{2} - 6 x + 9) \end{aligned}$

:a+b)(a2-ab+b2)=(2x+3)((2x)2-(2x)(3)+32)=(2x+3)(4x2-6x+9)

Therefore , $\begin{aligned} 8 x^{3} + 27 = (2 x + 3) (4 x^{2} - 6 x + 9) \end{aligned}$ . The 2nd factor is prime, so this is factored completely.
::因此,8x3+27=(2x+3)(4x2-6x+9),第二个因数为主要因数,因此完全计入。

This video by CK-12 demonstrates how to factor the sum of cubes.
::CK-12的这段视频展示了如何乘以立方体的总和。

Example 2
::例2

Factor $\begin{aligned} x^{3} - 125 \end{aligned}$ .
::系数 x3-125.

Solution: Factor using the difference of cubes formula , where $\begin{aligned} a = x \end{aligned}$ and $\begin{aligned} b = 5 \end{aligned}$ .
::解析度:使用立方体公式差异的系数,其中a=x和b=5。

$\begin{aligned} x^{3} - 125 & = (x - 5) ((x)^{2} + 5 x + (5)^{2}) \\ = (x - 5) (x^{2} + 5 x + 25) \end{aligned}$

::x3-125=(x-5)((x)2+5x+(5)2)=(x-5)(x2+5x+25)

This video by CK-12 demonstrates how to factor the difference of cubes.
::CK-12的这段视频展示了如何计算立方体的差别。

Example 3
::例3

Factor $\begin{aligned} x^{5} + 1000 x^{2} . \end{aligned}$
::系数 x5+1000x2。

Solution: First, take out any common factors.
::解决办法:首先,排除任何共同因素。

$\begin{aligned} x^{5} + 1000 x^{2} = x^{2} (x^{3} + 1000) \end{aligned}$

::x5+1000x2=x2(x3+1000)

What is inside the " data-term="Parentheses" role="term" tabindex="0"> parentheses is a sum of cubes. Using the formula, we have
::括号中括号中的内容是立方体的总和。使用公式,我们发现

$\begin{aligned} x^{5} + 1000 x^{2} & = x^{2} (x^{3} + 1000) \\ = x^{2} (x^{3} + 10^{3}) \\ = x^{2} (x + 10) (x^{2} - 10 x + 100) . \end{aligned}$

::x5+1000x2=x2(x3+100)=x2(x3+103)=x2(x+10)(x2-10x+100)。

Example 4
::例4

Say the amount of space, or volume, between the two walls in the Introduction can be written as $\begin{aligned} 729 - x^{3}, \end{aligned}$ where x is the length of the inner wall. Write this expression in factored form .
::说入门两面墙之间的空间或体积可以写为 729-x3, 其中x是内部墙的长度。以系数形式写下此表达式。

Solution: Factor using the difference of cubes formula, where $\begin{aligned} a = 9 \end{aligned}$ and $\begin{aligned} b = x \end{aligned}$ .
::解析度: 使用立方体公式差异的系数, 其中 a=9 和 b=x 。

$\begin{aligned} 729 - x^{3} & = 9^{3} - x^{3} \\ = (9 - x) (9^{2} + (9) (x) + (x)^{2}) \\ = (9 - x) (81 + 9 x + x^{2}) \end{aligned}$

::729-x3=93-x3=(9-x)(92+(9)(x)+(x)2)=(9-x)(81+9x+x2)

You can rewrite the 2nd factor as $\begin{aligned} x^{2} + 9 x + 81 \end{aligned}$ , but it is not necessary.
::您可以将第二个因数重写为 x2+9x+81, 但没有必要。

by Jennifer Caldwell demonstrates how to find the sum and difference of cubes.
::Jennifer Caldwell展示了如何找到立方体的总和和差异。

Summary
::摘要

To factor a sum of cubes, we have the form $\begin{aligned} a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2}) \end{aligned}$ .
::乘以立方体的总和,我们有表A3+b3=(a+b)(a2-ab+b2)。

To factor a difference of cubes, we have the form $\begin{aligned} a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2}) \end{aligned}$ .
::乘以差异的立方体,我们有表A3-b3=(a-b)(a2+ab+b2)。

Review
::回顾

Factor each polynomial by using the sum or difference of cubes.
::使用立方体的总和或差数乘以每个多数值。

1. $\begin{aligned} x^{3} - 27 \end{aligned}$
::1. x3 - 27

2. $\begin{aligned} 64 + x^{3} \end{aligned}$
::2. 64+x3

3. $\begin{aligned} 32 x^{3} - 4 \end{aligned}$
::3. 32x3-4

4. $\begin{aligned} 64 x^{3} + 343 \end{aligned}$
::4. 64x3+343

5. $\begin{aligned} 512 - 729 x^{3} \end{aligned}$
::5.512-729x3

6. $\begin{aligned} 125 x^{4} + 8 x \end{aligned}$
::6. 125x4+8x

7. $\begin{aligned} 648 x^{3} + 81 \end{aligned}$
::7. 648x3+81

8. $\begin{aligned} 5 x^{6} - 135 x^{3} \end{aligned}$
::8. 5x6-135x3

9. $\begin{aligned} 686 x^{7} - 1024 x^{4} \end{aligned}$
::9.866x7-1024x4

10. $\begin{aligned} 125 x^{3} + 1 \end{aligned}$
::10. 125x3+1

11. $\begin{aligned} 64 - 729 x^{3} \end{aligned}$
::11. 64-729x3

12. $\begin{aligned} 8 x^{4} - 343 x \end{aligned}$
::12. 8x4-343x

13. $\begin{aligned} 5 x^{5} + 625 x^{2} \end{aligned}$
::13. 5x5+625x2

14. $\begin{aligned} 686 x^{3} + 2000 \end{aligned}$
::14.86x3+2000

Explore More
::探索更多

1. You have a piece of cardboard that you would like to fold up and make into an open (no-top) box. The dimensions of the cardboard are $\begin{aligned} 36^{''} \times 42^{''} \end{aligned}$ . Write a factored equation for the volume of this box. Find the volume of the box when $\begin{aligned} x = 1, 3, \end{aligned}$ and 5.
::1. 您想要折叠一张纸板, 并制成一个打开的( 空顶) 框。纸板的尺寸是 3642。为此框的体积写一个计数方程式。在 x=1,3 和 5 时查找框的体积。

2. The volume of a rectangular prism is $\begin{aligned} 2 x^{4} - 128 x \end{aligned}$ . What are the lengths of the prism's sides?
::2. 矩形棱晶体积为2x4-128x。棱晶面的长度是多少?

3. In the next chapter, we will learn about an expression called the "discriminant," which can tell us what kind of factors we have. For a quadratic expression of the form $\begin{aligned} a x^{2} + b x + c, \end{aligned}$ the discriminant is $\begin{aligned} b^{2} - 4 a c \end{aligned}$ . Show that the 2nd polynomial factor in a sum or difference of two cubes, $\begin{aligned} x^{2} \pm m x + m^{2} \end{aligned}$ , is always prime by showing that the discriminant is always negative.
::3. 在下一章,我们将了解一个称为“差异性”的表达方式,它可以告诉我们我们有什么样的因素。对于Ax2+bx+c形式的二次表达方式来说,对立体是b2-4ac。显示以两个立方体(x2+bx+m2)的总和或差数为总和或差数的第二个多元系数,即 x2cmmx+m2, 显示对立方体总是负的, 总是最重要的。

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

计算立方体之和和之差

Sum and Difference of Two Cubes ::两个立方体的总和和和差额

Sum and Difference of Two Cubes ::两个立方体的总和和和差额

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

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

Sum and Difference of Two Cubes
::两个立方体的总和和和差额

Sum and Difference of Two Cubes
::两个立方体的总和和和差额

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

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