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我们来介绍

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多项式的余数定理

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当我们往下看的时候

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一开始您可能觉得有点神奇

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但是在今后的视频中，我们会去证明它

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然后我们就明白，就像数学中的其他很多东西

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当您真正理解的时候

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或许它就不会显得那么神奇了

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那么什么是多项式的余数定理呢

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题目告诉我们，如果

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有一个多项式函数，f(x)

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这就是这个多项式

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多项式

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它来除以

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（x-a)

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那么余数

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从那个多项式

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的长除法我们就会得到答案为f(a)

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其结果将是

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f(a)

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我想这个看起来有点抽象

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我指的是f(x)以及f(x-a)

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我来把它说得具体一些

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我们来算f(x)等于多少

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我们来假设一个

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2次幂的多项式

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其实对所有的多项式其实都是成立的

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所以3x的平方

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减4x加7

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我们假设a等于，a等于1

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所以我们的除法就是

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变成我们要除以

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（x-1）

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所以a，在这个例子中，等于1

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现在来做多项式的长除法

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我建议您暂停视频

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如果您不熟悉长除法

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我建议您先去看之前那个视频

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因为我假设您已经

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知道怎么去做多项式的长除法

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3x^-4x+7

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除以(x-1)

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来计算余数是多少

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并且来看余数是不是等于f(1)

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假设您已经试过了

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我们现在来一起做

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我们的除数为(x-1)

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被除数为3x^2

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减4x+7

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好了，做一点多项式的长除法

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来开启您的一天是不错的方法

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我就是这样的

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我不知道您的早上以什么开始

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好了，我们来看x项

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x的最高幂次项

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我从最高幂次项开始

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那么x乘以什么得到3x的平方

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3x的平方是多少

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3x的平方就是3乘以x的平方

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所以我可以写下3x

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我可以写在

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x的一次方的位置

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3x乘以x是3x的平方

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3x乘以负1等于负3x

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现在来做减法

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这个和您通常做的长除法是一回事

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那么结果是什么

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3x的平方减3x的平方

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等于0

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这项相抵减

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这一项-4x

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要加3x,对吧

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因为负负得正

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负4x加3x

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得到负x

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我用新的颜色来做

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我们得到负x

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然后把7落下

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将您第一次学习长除法和这个做一个完全的比较

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也许您在3年级或者4年级学过

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我做的就是用3x来乘以这个

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您就得到3x平方减3x

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然后我再来做减法，被减数为3x平方

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减4x，然后就得到这个

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或者您可以说我用整个多项式

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来减除数然后就得到-x+7

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现在，-x+7里面有多少

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（x-1)呢

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x被负x除

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商负1，乘以x

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就是负x

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负1乘以负1等于正1

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然后我们又要来做减法

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我们要来减去这个

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然后我们就得到余数了

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所以负x减去负x

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就等于负x加x

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它们相加等于0

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现在落下7

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然后7加1

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不要忘记括弧外面有负号

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所以如果您将负号分配进来

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那就等于负1

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7减去1等于6

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所以您的余数为6

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一种方法去想

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您可以说,算了

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我还是留给将来的视频课来解释

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这个就是余数

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您知道当您到余数这一步的时候

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这就是对长除法做了一个复习

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也就是当您得到一个低次幂的时候

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在这里，我想您可以这么来叫它

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是一个零次幂的多项式

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这个比您实际上

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要除以的因子(x-1)的x次幂要低

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因为它比因子的幂次低，所以它就是余数了

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您不能再用这个余数去商任何数

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现在，就这个多项式余数定理

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如果它是正确的，因为我在这里只是任意选了一个例子而言

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这不是一个证明，而是仅仅是

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一种直观的方法

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让我们知道这个余数定理

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如果这个余数定理是正确的

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它告诉我们的是，在这个实例中

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f(1)应该等于6

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它应该等于这个余数

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现在来看看对不对

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它应该等于3乘以1的平方

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等于3，减4乘以1

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那么就是减4，再加7

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3减4等于负1再加7，的确是

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我们被其结果所鼓舞

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它真的就等于6

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所以，至少就这个特殊的例子而言

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看起来是正确的，它

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证明余数定理是对的

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但是这个定理的作用是，如果有人问

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"如果我用3x^-4x+7 除以

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(x-1) 的话，如果我只关心余数

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那么余数是什么？“

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这里他们并不关心实际的商会是多少

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他们就只想知道余数，您可以说

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“你们看，在这个例子里，因为a是1

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我可以就把1代入

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我可以算出f(1),然后我得出答案是6

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我并不需要去做这个整个长除法过程

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我仅仅只需要去做这个步骤

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就可以算出余数“

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也就是得出3x的平方减4x加7

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除以x-1的余数