3??11???2?x1????y1?1??????x?01?y, ?2????2?2?x?????3?001??y3???????2化为g(x1,x2,x3)?y12?4y2?(y1?2y2)(y1?2y2). 由Y?P?1X, 得y1?x1?x2?x3,
y2?x2?1x3, y3?x3. 于是g(x1,x2,x3)?(x1?x2?2x3)(x1?3x2). 2故f(x1,x2)?g(x1,x2,1)?(x1?x2?2)(x1?3x2).
2例10 多项式f(x1,x2)?x12?2x2?22x1x2?6x1?62x2?9在R上能否分解? 如果
能,将其分解．
22解 考虑二次型g(x1,x2,x3)?x12?2x2?9x3?22x1x2?6x1x3?62x2x3, 其矩阵为
?1?A???2??3??22?323??1?23?????32???000?
??000?9????则秩rankA?1, 由定理2.6知, g(x1,x2,x3)能在R上分解, 则f(x1,x2)?g(x1,x2,1)也能在R上分解. 易得
f(x1,x2)?g(x1,x2,1)?(x1?2x2?3)2.