定義
我們把形如
![整係數多項式](/img/5/9a8/wZwpmL1MDN3kzMxkjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL5YzLyQzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/5/439/wZwpmL1EDOxEDO2UTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1UzL2EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/6/8b3/wZwpmL4MDN4kTMyUDM0kTO0UTMyITNykTO0EDMwAjMwUzL1AzL3czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/a/9d6/wZwpmL1IDOzQjM1UjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1YzL1MzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/1/fdf/wZwpmLyAzNzYjMyYDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL2gzL1IzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/262/wZwpmLzYjM1ITMwETN0YzM1UTM1QDN5MjM5ADMwAjMwUzLxUzL4QzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/f/578/wZwpmLycDNwEjNykzN5ADN0UTMyITNykTO0EDMwAjMwUzL5czL1MzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/b/e53/wZwpmLxczN3YjMyUTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1UzL4gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
的表達式叫做x的多項式,記為 ,其中n是正整數,x是一個符號(或文字), 都是常數,叫做 的 係數, 還叫做 的 常數項。 叫做 一個項,k叫做這一項的 次數。當 時, 叫做 的 首 項, 叫做 首項係數,n叫做 的次數,記為次 .如果 的係數全為零,則把 叫做 零多項式,記為0。我們認為零多項式沒有次數,若 ,則說 是零次多項式。
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/5/439/wZwpmL1EDOxEDO2UTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1UzL2EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
如果多項 的係數 都是整數,我們把 叫做 整係數多項式,如果 的係數都是有理數,就把 叫做 有理係數多項式。同樣地,可以定義 實係數 多項式和 復係數多項式。
相關性質
定理1
![整係數多項式](/img/6/a14/wZwpmL2YTOxAjM3IDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzLxIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![整係數多項式](/img/4/953/wZwpmL3IDO0kTMxkjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL5YzL3AzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
(整係數多項式有理根的性質) 若既約分數 為整係數多項式 的根,則:
![整係數多項式](/img/7/48e/wZwpmL1QzNygTM3UTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1UzL0czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![整係數多項式](/img/e/a44/wZwpmL1YTN4gjM0IjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyYzLxgzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
(1) 且 ;
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/1/3d3/wZwpmLxgTM3YTM3IDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(2) 除以 所得的商的各項係數必為p的倍數。
![整係數多項式](/img/6/b36/wZwpmLzEjNwQjM5ETN0YzM1UTM1QDN5MjM5ADMwAjMwUzLxUzLxUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
證明:因為 為 的根。
所以,由因式定理,有
![整係數多項式](/img/0/9a5/wZwpmL1EzM4YDOzcjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL3YzLwgzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
其中
![整係數多項式](/img/2/15d/wZwpmLwUjNzkzMzIDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzL2YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
為有係數多項式。由此可得
![整係數多項式](/img/4/b83/wZwpmL3MTOyYDOwIjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyYzLzIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
可化為
![整係數多項式](/img/5/466/wZwpmLxIDMwQzN1gzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL4czLyEzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
由根的意義,有
![整係數多項式](/img/8/999/wZwpmL3MDO4czM5gzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL4czLzczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
可化為
![整係數多項式](/img/1/584/wZwpmLzYTN3gDOwIjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyYzL3EzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
故得
![整係數多項式](/img/4/5cd/wZwpmL2YTOwUDMykTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL5UzL4czLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
①
![整係數多項式](/img/d/dbd/wZwpmL4AzM4cDM4EDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLxgzL1czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![整係數多項式](/img/c/323/wZwpmLzgjMxQjN1kjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL5YzL2YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![整係數多項式](/img/1/f63/wZwpmL0YDN2QDN3IjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyYzL0AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/b/f95/wZwpmL2ATN0MzN4QzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0czL1QzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
可知 為整數,但 ,故 為整數,又 時,式①變為
![整係數多項式](/img/d/854/wZwpmL2ETN2cDO3QTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0UzL0czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
由此得
![整係數多項式](/img/5/b28/wZwpmL2UTO1YDM5IjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyYzL3UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
推論1
最高次項係數為1的整係數多項式的有理根必為整數。
推論2
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/69e/wZwpmL1AjN2ITN4QjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0YzLwIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/6/442/wZwpmL3cTNykzNxMTN0YzM1UTM1QDN5MjM5ADMwAjMwUzLzUzLwgzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
若既約分數 為整係數多項式 的根,則除以 所得的商為整係數多項式。
定理2
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/6/a14/wZwpmL2YTOxAjM3IDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzLxIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/1/3d3/wZwpmLxgTM3YTM3IDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![整係數多項式](/img/1/3d3/wZwpmLxgTM3YTM3IDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/6/a14/wZwpmL2YTOxAjM3IDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzLxIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![整係數多項式](/img/6/a14/wZwpmL2YTOxAjM3IDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzLxIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
求整係數多項式 的有理根的主要依據,其方法是:首先按定理1的結論(1)或其他有關條件找出有理根的一切可能根 ,然後採用綜合除法將 除以 ,根據因式定理,若且唯若 整除 時, 為有理根,除的過程中,如發現商的係數非p的倍數,則根據定理1的結論(2), 非有理根,可不必再除下去。
![整係數多項式](/img/4/5ca/wZwpmL4QTO3gTM4UjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1YzL3czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![整係數多項式](/img/5/038/wZwpmLxgzNyADOxQDO0YzM1UTM1QDN5MjM5ADMwAjMwUzL0gzL1AzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
例1設,求證:為無理數。
![整係數多項式](/img/5/038/wZwpmLxgzNyADOxQDO0YzM1UTM1QDN5MjM5ADMwAjMwUzL0gzL1AzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/204/wZwpmLxYDMwAzNzUjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1YzL4AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
證明:是整係數多項式的一根,
![整係數多項式](/img/3/204/wZwpmLxYDMwAzNzUjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1YzL4AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
可能的有理根為±1,±2,但
![整係數多項式](/img/9/abb/wZwpmL4ITMzQjN1IzN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyczL0QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
所以無有理數根。
定理3
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/d/375/wZwpmL1AzNxEjM5IjM5kzM0UTMyITNykTO0EDMwAjMwUzLyIzL2gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/d/9ca/wZwpmLyMjM2ATNwIjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyYzLzYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
設 為有理係數多項式, 與 有公共根a,則。
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/d/375/wZwpmL1AzNxEjM5IjM5kzM0UTMyITNykTO0EDMwAjMwUzLyIzL2gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/d/375/wZwpmL1AzNxEjM5IjM5kzM0UTMyITNykTO0EDMwAjMwUzLyIzL2gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
定理3的意義是,若有理係數不可約多項式有一根為另一有理係數多項式 的根,則 的全部根均為 的根。
推論3
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/d/1fd/wZwpmL4YDOzQTM3gTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL4UzL4AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![整係數多項式](/img/0/a69/wZwpmL0ATN2cDM1gjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL4YzL4YzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![整係數多項式](/img/8/bc7/wZwpmL3ADOygjM0cTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL3UzLzEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/e/098/wZwpmL3IDMxcDN1UzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL1czL0UzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![整係數多項式](/img/7/72e/wZwpmL4cjM2cTNzUjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1YzL0EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
如果有理係數多項式 有無理根 ( , 為無理數, ),則必有無理根。
推論4
![整係數多項式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/f/8ac/wZwpmL4gTM5ETNzIjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyYzLxYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![整係數多項式](/img/9/51a/wZwpmLzETN4cDN4ADO0YzM1UTM1QDN5MjM5ADMwAjMwUzLwgzL4gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![整係數多項式](/img/d/53b/wZwpmLzETNyYTO5IjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyYzL4IzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![整係數多項式](/img/d/50f/wZwpmL4EzM5gTM4QzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0czLzYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
如果有理係數多項式 有無理根 ( 為無理數且非同類根式),則必有無理根 及。