Section: 长长的多配偶分会 | 6.6 长期一夫多妻制

Section outline

Engineers often use to model the path of a roller coaster. If we can find a line that just touches a graph at a point, we can estimate the speed of the roller coaster at that moment in time. In this section, we discuss , which will help us find this line.
::工程师们经常使用过山车的路径模型。如果我们能找到一条在某个点仅触摸一个图形的线条, 我们可以估计一下当时的过山车的速度。在这一部分, 我们讨论一下, 这将帮助我们找到这条线。

Division of Polynomials
::多边分会

Division of polynomials is similar to division of numbers. When we divide numbers, we divide into the largest place value first. Analogously, for polynomials, we divide into the largest degree first.
::多元数的除法与数字的除法相似。当我们分数时, 我们先将数字分成最大的位值。类似地, 对于多元数, 我们先将数字分为最大的位值。

Let's consider an example.
::让我们举个例子。

Example 1
::例1

Find the quotient of $\begin{aligned} (2 x^{3} - 3 x^{2} - 8 x + 12) \div (2 x - 3) \end{aligned}$ .
::查找( 2x3- 3x2- 8x+12) ( 2x- 3) 的商数。

Solution: First, put the dividend , what we are dividing, into $\begin{aligned} 2 x^{3} - 3 x^{2} - 8 x + 12 \end{aligned}$ , inside the division bar and put the divisor , what we are dividing by, $\begin{aligned} 2 x - 3 \end{aligned}$ , outside.
::解决方案:第一,将红利,我们所分割的, 分为2x3-3x2-8x+12, 放在分局的酒吧里, 把分局, 我们所分割的, 2x-3, 外面。

$\begin{aligned} 2 x - 3 2 x^{3} - 3 x^{2} - 8 x + 12 \end{aligned}$

::2 - 32x3 - 3x2 - 8x+12

We start with the largest power terms $\begin{aligned} 2 x \end{aligned}$ and $\begin{aligned} 2 x^{3} \end{aligned}$ . The question is what do we multiply $\begin{aligned} 2 x \end{aligned}$ by to get $\begin{aligned} 2 x^{3} \end{aligned}$ : $\begin{aligned} x^{2} \end{aligned}$ .
::我们从最大功率值 2x 和 2x3 开始。问题是我们乘以 2x 乘以 2x 乘以 2x 3: x2 。

$\begin{aligned} x^{2} \\ 2 x - 3 2 x^{3} - 3 x^{2} - 8 x + 12 \\ \underline{2 x^{3} - 3 x^{2}} \\ 0 \end{aligned}$

::x22x - 32x3 - 3x2 - 8x+12 2x3 - 3x2_ 0

Place $\begin{aligned} x^{2} \end{aligned}$   above the $\begin{aligned} x^{3} \end{aligned}$ term in the polynomial .
::加号中的 x3 术语上方的 x2 位。

Multiply $\begin{aligned} x^{2} \end{aligned}$ by both terms in the divisor and place them until their like terms . Subtract from the dividend. Pull down the next two terms and repeat.
::将 x2 乘以两个条件, 并在 divisor 中乘以两个条件, 并将其放置到相同的条件中。从红利中扣除。将接下来的两个条件拖下水并重复。

$\begin{aligned} x^{2} - 4 \\ 2 x - 3 2 x^{3} - 3 x^{2} - 8 x + 12 \\ \underline{2 x^{3} - 3 x^{2}} \\ - 8 x + 12 \\ \underline{- 8 x + 12} \end{aligned}$

::x2-42x - 32x3 - 3x2 - 8x+12 2x3 - 3x2_ - 8x12 - 8x12

Next, we can multiply $\begin{aligned} 2 x \end{aligned}$ by -4 to get the new highest power term, $\begin{aligned} - 8 x \end{aligned}$ .
::接下来,我们可以乘以 2x 乘以 -4 获得新的最高权力术语, - 8x 。

After multiplying both terms in the divisor by -4, place that under the terms that were brought down. When subtracting we notice that everything cancels out. There is no remainder.
::将两个条件乘以 4 后, 将两个条件都乘以 4 , 将这两个条件置于被降的条件之下。当减去时, 我们注意到一切都取消了。没有剩余了。

by CK-12 demonstrates how to perform polynomial long division .
::CK-12 演示如何进行多面长的分割。

Example 2
::例2

Find the quotient of $\begin{aligned} (2 x^{3} - 6 x^{2} + 5 x - 20) \div (x^{2} - 5) \end{aligned}$ .
::查找( 2x3- 6x2+5x- 20) ( x2- 5) 的商数。

Solution: Set up the problem using a long division bar.
::解决方案: 使用长分隔栏设置问题。

$\begin{aligned} x^{2} - 5 2 x^{3} - 6 x^{2} + 5 x - 20 \end{aligned}$

::x2 - 52x3 - 6x2+5x-20

What do we multiply $\begin{aligned} x^{2} \end{aligned}$ by to get $\begin{aligned} 2 x^{3} \end{aligned}$ ? That would be $\begin{aligned} 2 x \end{aligned}$ .
::我们乘 x2 乘以什么才能得到 2x3?

$\begin{aligned} 2 x \\ x^{2} - 5 2 x^{3} - 6 x^{2} + 5 x - 20 \\ \underline{2 x^{3} - 10 x^{2}} \\ 4 x^{2} + 5 x - 20 \end{aligned}$

::2x2 - 52x3 - 6x2+5x-2 - 20 2x3 - 10x2 - 4x2+5x-20

Multiply $\begin{aligned} 2 x \end{aligned}$ by the divisor. Subtract that from the dividend.
::乘以2x乘以分数,减去股息

Repeat the previous steps with the new highest power term, $\begin{aligned} 4 x^{2} \end{aligned}$ . Now, what do we multiply $\begin{aligned} x^{2} \end{aligned}$ by to get $\begin{aligned} 4 x^{2} \end{aligned}$ ? That would be 4.
::重复前几个步骤, 新的最高功率术语是 4x2 。现在, 我们乘 x2 乘以 4x2 来取得 4x2 是什么? 这将是 4 。

$\begin{aligned} 2 x + 4 \\ x^{2} - 5 2 x^{3} - 6 x^{2} + 5 x - 20 \\ \underline{2 x^{3} - 10 x^{2}} \\ 4 x^{2} + 5 x - 20 \\ \underline{4 x^{2} - 20} \\ 5 x \end{aligned}$

::2x+4x2 - 52x3 - 6x3 - 6x2 - 5x2 - 20 2x3 - 10x2 - 4x2 - 4x2 - 5x - 20 4x2 - 20_ 5x

At this point, we are done. $\begin{aligned} x^{2} \end{aligned}$ cannot go into $\begin{aligned} 5 x \end{aligned}$ because it has a higher degree. Therefore , $\begin{aligned} 5 x \end{aligned}$ is a remainder. The complete result would be $\begin{aligned} 2 x + 4 + \frac{5 x}{x^{2} - 5} \end{aligned}$ .
::此时,我们就完成了。 x2 不能进入 5x, 因为它具有更高的学位。因此, 5x 是一个剩余部分。完整的结果将是 2x+4+5x2-5 。

WARNING
::警告

One of the most common problems is a problem of bookkeeping.
::最常见的问题之一是簿记问题。

Make sure to line up the term you are adding to the quotient over the term you divided into. Otherwise, you can forget to divide into some terms in the dividend.
::确保您在您所划分的术语的商数中添加的词行一致。否则,您可以忘记在红利中将它分为某些术语。

Example 3
::例3

Find the quotient of $\begin{aligned} (3 x^{4} + x^{3} - 17 x^{2} + 19 x - 6) \div (x^{2} - 2 x + 1) \end{aligned}$ to determine if $\begin{aligned} x^{2} - 2 x + 1 \end{aligned}$ goes evenly into $\begin{aligned} 3 x^{4} + x^{3} - 17 x^{2} + 19 x - 6 \end{aligned}$ .
::查找( 3x4+x3- 17x2+19x- 6) (x2-2x+1) 的商数以确定 x2-2x+1 是否均匀地进入 3x4+x3- 17x2+19x-6 。

Solution: First, do the long division. If $\begin{aligned} x^{2} - 2 x + 1 \end{aligned}$ goes in evenly, then the remainder will be zero.
::解决方案 : 首先, 执行长的分隔。如果 x2 - 2x+1 平均进入, 那么剩余部分将是零。

$\begin{aligned} 3 x^{2} + 7 x - 6 \\ x^{2} - 2 x + 1 3 x^{4} + x^{3} - 17 x^{2} + 19 x - 6 \\ \underline{3 x^{4} - 6 x^{3} + 3 x^{2}} \\ 7 x^{3} - 20 x^{2} + 19 x \\ \underline{7 x^{3} - 14 x^{2} + 7 x} \\ - 6 x^{2} + 12 x - 6 \\ \underline{- 6 x^{2} + 12 x - 6} \\ 0 \end{aligned}$

::3x2+7x - 6x2 - 2x - 6x2 - 2x2 - 6x2 - 6x2 - 13x4+3x3+3x2_ 3x4 - 6x2 - 6x3+3x2_ 7x3 - 202+19x7x3 - 14x2+7x_ - 6x2+12x6 - 6x2+12x6_ 0

This means that $\begin{aligned} x^{2} - 2 x + 1 \end{aligned}$ and $\begin{aligned} 3 x^{2} + 7 x - 6 \end{aligned}$ both go evenly into $\begin{aligned} 3 x^{4} + x^{3} - 17 x^{2} + 19 x - 6 \end{aligned}$ .
::这意味着 x2 - 2x+1 和 3x2+7x-6 均匀地进入 3x4+x3 - 17x2+19x-6。

by Mathispower4u demonstrates how to divide a polynomial by a binomial using long division.
::由Mathispower4u 演示如何用长期的二进制来划分多式。

Example 4
::例4

The equation that models part of a roller coaster is $\begin{aligned} y = - x^{3} + 5 x^{2} - 4 x \end{aligned}$ , where x is seconds times 10 ( $\begin{aligned} 10 x \end{aligned}$ seconds) and y is feet times 10 ( $\begin{aligned} 10 y \end{aligned}$ feet). Determine the speed of the roller coaster at $\begin{aligned} x = 4 \end{aligned}$ .
::云层滑雪车的模型部分方程式是yx3+5x2-4x,其中x乘以秒乘以10(10x秒),y为英尺乘以10(10y英尺)。确定乘以x=4的云层滑雪车速度。

Solution: The speed of the roller coaster is the of the tangent line, the line that touches the graph at one point, in this case at $\begin{aligned} x = 4 \end{aligned}$ . We can find this by dividing $\begin{aligned} - x^{3} + 5 x^{2} - 4 x \end{aligned}$ by $\begin{aligned} (x - 4)^{2} = x^{2} - 8 x + 16 \end{aligned}$ . The remainder will be the tangent line.
::解析度: 过山车的速率是正切线的速率, 这条线在某个点上触动图形, 在 x=4 的情况下, 在 x=4 。我们通过将 - x3+5x2 - 4x 除以 (x-4) 2 = x2 - 8x+16 来找到这个速率。剩下的将是正切线。

$\begin{aligned} - x - 3 \\ x^{2} - 8 x + 16) \bar{- x^{3} + 5 x^{2} - 4 x} \\ \underline{- x^{3} + 8 x^{2} - 16 x} \\ - 3 x^{2} + 12 x \\ \underline{- 3 x^{2} + 24 x - 48} \\ - 12 x + 48 \end{aligned}$

::- 3x2 - 8x+16 - x3+5x2 - 4x *x3+8x2 - 16x 3x2 - 12x2 - 3x2 - 3x2+24x - 48* 12x+48

Our remainder is $\begin{aligned} - 12 x + 48 \end{aligned}$ . That is the equation of the tangent line at $\begin{aligned} x = 4 \end{aligned}$ : $\begin{aligned} y = - 12 x + 48 \end{aligned}$ . The slope is -12, so the roller coaster is going down at a rate of 12 feet per second.
::我们的剩余部分是 - 12x+48。这是 x=4 的正切线的方程式 : y12x+48。斜坡是 - 12, 因此过山车以每秒12英尺的速度下降。

Summary
::摘要

To divide polynomials, approach a problem similar to dividing numbers. However, instead of using the highest place value, we use the highest degree term.
::要分割多种族, 请处理一个类似于分隔数字的问题。但是, 我们不是使用最高位值, 而是使用最高位值。

Review
::回顾

Divide the following polynomials using long division.
::使用长除法除以下列多数值。

1. $\begin{aligned} (2 x^{3} + 5 x^{2} - 7 x - 6) \div (x + 1) \end{aligned}$
::1. (2x3+5x2-7x-6)(x+1)

2. $\begin{aligned} (x^{4} - 10 x^{3} + 15 x - 30) \div (x - 5) \end{aligned}$
::2. (x4-10x3+15x-30) (x-5)

3. $\begin{aligned} (2 x^{4} - 8 x^{3} + 4 x^{2} - 11 x - 1) \div (x^{2} - 1) \end{aligned}$
::3. (2x4-8x3+4x2-11x-1)(x2-1)

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

5. $\begin{aligned} (3 x^{4} - 5 x^{3} - 21 x^{2} - 30 x + 8) \div (x - 4) \end{aligned}$
::5. (3x4-5x3-21x2-30x+8)(x-4)

6. $\begin{aligned} (2 x^{5} - 5 x^{3} + 6 x^{2} - 15 x + 20) \div (2 x^{2} + 3) \end{aligned}$
::6. (2x5-5x3+6x2-15x+20)(2x2+3)

7. $\begin{aligned} x^{3} - 9 x^{2} + 27 x - 15; (x + 5) \end{aligned}$
::7. x3-9x2+27x-15; (x+5)

8. $\begin{aligned} x^{3} + 4 x^{2} - 9 x - 36; (x + 4) \end{aligned}$
::8. x3+4x2-9x-36; (x+4)

9. $\begin{aligned} 2 x^{3} + 7 x^{2} - 7 x - 30; (x - 2) \end{aligned}$
::9. 2x3+7x2-7x2-7x-30;(x-2)

10. $\begin{aligned} (5 x^{4} + 6 x^{3} - 12 x^{2} - 3) \div (x^{2} + 3) \end{aligned}$
::10. (5x4+6x3-12x2-3)(x2+3)

Explore More
::探索更多

1. The idea behind Euclidean Division is that a function (dividend) equals a divisor times the quotient plus the remainder. Or,
::1. Euclidean司背后的想法是,函数(dividd)等于商数乘以差数加上剩余数。

$\begin{aligned} a = b q + r & a : dividend \\ b : divisor \\ q : quotient \\ r : remainder \end{aligned}$
Express $\begin{aligned} x^{4} + 3 x^{3} - 8 x^{2} - 9 \end{aligned}$ in this form when dividing by $\begin{aligned} x - 2 \end{aligned}$ .
::a=bq+ra:divididdb:divisorq:quotientr:remainder Express x4+3x3-8x2-9, 以这种形式除以 x-2 。

2. The Remainder Theorem says that the remainder from dividing a polynomial function, $\begin{aligned} f (x) \end{aligned}$ , by $\begin{aligned} x - c \end{aligned}$ is $\begin{aligned} f (c) \end{aligned}$ . Confirm this in an example by dividing and evaluating the function $\begin{aligned} f (x) = 4 x^{3} - 6 x^{2} - 7 x + 5 \end{aligned}$ when $\begin{aligned} c = 2 \end{aligned}$ .
::2. 维护者理论论称,从将多面函数f(x)除以x-c除以x(x)至x-c的剩余部分为 f(c) 。在一个例子中确认这一点,在 c=2时分割和评估函数f(x)=4x3-6x2-7x+5。

3. Find the tangent line to the graph of $\begin{aligned} y = x^{3} - 4 x^{2} + 9 x - 13 \end{aligned}$ at $\begin{aligned} x = 1 \end{aligned}$ by dividing by $\begin{aligned} (x - 1)^{2} \end{aligned}$ .
::3. 在 x=1 时查找 y=x3 - 4x2+9x- 13 图形的正切线,除以 (x-1) 2 。

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

Division of Polynomials ::多边分会

Example 1 ::例1

Example 2 ::例2

WARNING ::警告

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

Division of Polynomials
::多边分会

Example 1
::例1

Example 2
::例2

WARNING
::警告

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