怎么用综合除法给多项式做除法运算

综合除法是一种除多项式的快速方法，其中需要除以多项式的系数。除去其中的变量和指数。这种方法和普通的长除法方式不同，是将除得的数加起来，而不是减掉。下面介绍给你综合除法的步骤。

步骤

1. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/7\/7c\/Divide-Polynomials-Using-Synthetic-Division-Step-1-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-1-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/7\/7c\/Divide-Polynomials-Using-Synthetic-Division-Step-1-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-1-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 1 写下问题。本例子中，要让x³ + 2x² - 4x + 8 除以 x + 2。 在分子位置写下第一个多项式（被除数），分母写下除数。
2. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/d\/d1\/Divide-Polynomials-Using-Synthetic-Division-Step-2-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-2-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/d\/d1\/Divide-Polynomials-Using-Synthetic-Division-Step-2-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-2-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 2 把除数中常数项符号倒转。 x + 2的常数项是2， 把它变为 -2
3. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/e\/ed\/Divide-Polynomials-Using-Synthetic-Division-Step-3-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-3-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/e\/ed\/Divide-Polynomials-Using-Synthetic-Division-Step-3-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-3-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 3 把这个数字写在倒置的除法运算符外边，如图。倒过来的除法运算符是一个倒转的L型。把-2放在左边。
4. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/4\/43\/Divide-Polynomials-Using-Synthetic-Division-Step-4-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-4-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/4\/43\/Divide-Polynomials-Using-Synthetic-Division-Step-4-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-4-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 4 把所有被除数的系数放在除法运算符里面。按原来顺序从左到右写，如 -2| 1 2 -4 8
5. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/9\/9f\/Divide-Polynomials-Using-Synthetic-Division-Step-5-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-5-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/9\/9f\/Divide-Polynomials-Using-Synthetic-Division-Step-5-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-5-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 5 把第一个系数移下去。把第一个系数1移下去。看起来是：
   • -2| 1  2  -4  8    ↓    1（文字显示有误的话，按图来写）
6. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/4\/4a\/Divide-Polynomials-Using-Synthetic-Division-Step-6-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-6-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/4\/4a\/Divide-Polynomials-Using-Synthetic-Division-Step-6-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-6-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 6 第一个系数乘以除法运算符上的数字（这里的除数），放在第二个系数下方。只要把1乘以-2得到-2，写在2下方就行:
   • -2| 1  2  -4  8        -2    1
7. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/3\/31\/Divide-Polynomials-Using-Synthetic-Division-Step-7-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-7-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/3\/31\/Divide-Polynomials-Using-Synthetic-Division-Step-7-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-7-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 7 把第二个系数和积加起来，把答案写在下面。现在把第二个系数2，加上刚刚得到的积-2，得到0。把这个数字放在上两个数字之下，和长除法类似:
   • -2| 1  2  -4  8        -2    1   0（按图来写）
8. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/7\/7f\/Divide-Polynomials-Using-Synthetic-Division-Step-8-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-8-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/7\/7f\/Divide-Polynomials-Using-Synthetic-Division-Step-8-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-8-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 8 把这个和再乘以除数，再放在第三个系数下。现在和是0，乘以除数-2，还是0。放在-4下面，得到：
   • -2| 1  2  -4  8        -2  0     1  （按图来写）
9. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/f\/f4\/Divide-Polynomials-Using-Synthetic-Division-Step-9-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-9-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/f\/f4\/Divide-Polynomials-Using-Synthetic-Division-Step-9-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-9-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 9 把积和第三个系数加起来，在积下方写下结果。把0加-4得-4，写在0下方:
   • -2| 1  2  -4  8        -2   0     1   0   -4 （按图来写）
10. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/5\/57\/Divide-Polynomials-Using-Synthetic-Division-Step-10-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-10-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/5\/57\/Divide-Polynomials-Using-Synthetic-Division-Step-10-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-10-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 10 这个数字再乘以除数，写在最后一个系数下，再加上系数。现在-4乘-2是8，放在第四个系数下。加系数得到 8 + 8 = 16。因此这是余数。写在积的下面：
    • -2| 1  2  -4  8        -2   0   8    1   0   -4   |16（按图来写）
11. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/f\/fb\/Divide-Polynomials-Using-Synthetic-Division-Step-11-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-11-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/f\/fb\/Divide-Polynomials-Using-Synthetic-Division-Step-11-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-11-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 11 把每个系数旁边放个变量，变量次数比原系数旁的变量次数都小1。本例中，第一个和，1放在x二次方边，第二个0放在x旁（可消掉此项），第三个系数就变成了常数项。然后在16旁边写个R，代表这是余数：
    • -2| 1  2  -4  8        -2   0   8    1   0   -4   |16    x²   + 0x    - 4    R 16 x² - 4 R16 （按图）
12. {"smallUrl":"https:\/\/www.zenmeban.com\/images_en\/thumb\/c\/c9\/Divide-Polynomials-Using-Synthetic-Division-Step-12-Version-2.jpg\/v4-460px-Divide-Polynomials-Using-Synthetic-Division-Step-12-Version-2.jpg","bigUrl":"https:\/\/www.zenmeban.com\/images\/thumb\/c\/c9\/Divide-Polynomials-Using-Synthetic-Division-Step-12-Version-2.jpg\/v4-728px-Divide-Polynomials-Using-Synthetic-Division-Step-12-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><\/div>"} 12 写下最终答案。最终答案是新的多项式x² - 4，加上余数 16 乘以x + 2的积: x² - 4 +16/(x +2)（按图）

小提示

• 要验证答案，把除数乘以刚刚得到的商，加上余数，应该和原式一样。

  (除数)(商)+(余数)

  (x + 2)(x² - 4) + 16

  用FOIL方法（First, Outer, Inner, Last-这是多项式相乘的一种顺序，或叫首项相乘，外项相乘，内项相乘，次项相乘）算出最后的多项式

  (x³ - 4x + 2x² - 8) + 16

  x³ + 2x² - 4x - 8 + 16

  x³ + 2x² - 4x + 8