解题方法
1 . 我们把
(其中
,
)称为一元n次多项式方程.代数基本定理:任何复系数一元
次多项式方程(即
,
,
,…,
为实数)在复数集内至少有一个复数根;由此推得,任何复系数一元
次多项式方程在复数集内有且仅有n个复数根(重根按重数计算).那么我们由代数基本定理可知:任何复系数一元
次多项式在复数集内一定可以分解因式,转化为n个一元一次多项式的积.即
,其中k,
,
,
,
,……,
为方程
的根.进一步可以推出:在实系数范围内(即
,
,
,…,
为实数),方程
的有实数根,则多项式
必可分解因式.例如:观察可知,
是方程
的一个根,则
一定是多项式
的一个因式,即
,由待定系数法可知,
.
(1)解方程:
;
(2)设
,其中
,
,
,
,且
.
(i)分解因式:
;
(ii)记点
是
的图象与直线
在第一象限内离原点最近的交点.求证:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e138b0fc1c40ba1637098eb2a6efd580.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffa01f03fb074bff35b35e07047d11b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6368fec0c2c25db7c29b014d60270e97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f35f7dcce39f3d4dc6b7faf84dc1d0a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6368fec0c2c25db7c29b014d60270e97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10b328845a4b1881eee38084d5501224.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcde67e0b4461129e0c7e3a12df4634f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edffa0cf823fb77bb7e273db0e014743.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/483fd78fe6ed871ce859f4796ad7779c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/943b765718479c160ba61ec5c6f8c5f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e29bf5652f0d4f764c3606efcdb445f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3230af83e2c18650f1de0c88060c0b25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e138b0fc1c40ba1637098eb2a6efd580.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f35f7dcce39f3d4dc6b7faf84dc1d0a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e138b0fc1c40ba1637098eb2a6efd580.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf70f45c7f3a63a81001f87863f2c73c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2527822fd5ee6ded770ffc9857c41bff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b924d856924e8cf2b172d5cacffe0610.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f2c82aa40a712f2ef6fda7eaeb88a48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7344f58d5f08fab08d4e99baa13fa652.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd7126d6d76248996a222631cc9ea93c.png)
(1)解方程:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d58fc8760f5b4612d0f76133d938f4e9.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/536bbd87dd4193314aec2e214e5f05b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f35f7dcce39f3d4dc6b7faf84dc1d0a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1cdb8081eb1b3390b3730c01b9afb59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/653588ca473b428b4a437d6a8ed7a76c.png)
(i)分解因式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e42787c800e5f9c7ac483bea80d8440.png)
(ii)记点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7775aa57ca0e62216f3039ed88dceed0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c520c63104bb6669c3591bd100b10e1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51969fc1a8030cef11cab59267689e89.png)
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