Useful Algebraic Manipulations(有用的代数运算).pdf

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1、UsefulAlgebraicManipulationsandFactorizationsPResult1.-Differenceofsamepowersxn−yn=(x−y)n−1xn−1−iyifornon-negativeintegersnandallrealsi=0x,y.PResult2.-Sumofsameoddpowersx2n+1+y2n+1=(x+y)2n(−1)i(x2n−iyi)fornon-negativeintegersni=0andallrealsx,y.Result3

2、.-SpecialCaseofResult1.-Differenceofsquaresx2−y2=(x−y)(x+y)forallrealsx,y.Result4.-SpecialCaseofResult1.-Differenceofcubesx3−y3=(x−y)(x2+xy+y2)forallrealsx,y.Result5.-SumofSquares-Differentrepresentationsx2+y2=(x+y)2−2xy=(x−y)2+2xyforallrealsx,y.222Result6.

3、x2+y2+z2+xy+yz+zx=(x+y)+(y+z)+(z+x)forallrealsx,y,z.2222Result7.x2+y2+z2−xy−yz−zx=(x−y)+(y−z)+(z−x)forallrealsx,y,z.2Result8.x2+y2+z2+2(xy+yz+zx)=(x+y+z)2forallrealsx,y,z.Result9.x2+y2+z2+3(xy+yz+zx)=(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)forallrealsx,y,z.Resul

4、t10.xy+yz+zx−(x2+y2+z2)=(x−y)(y−z)+(y−z)(z−x)+(z−x)(x−y)forallrealsx,y,z.222Result11.x2+y2+xy=x+y+(x+y)forallrealsx,y.2222Result12.x2+y2−xy=x+y+(x−y)forallrealsx,y.2Result13.4(x2+xy+y2)=3(x+y)2+(x−y)2forallrealsx,yResult14.3(x2−xy+y2)=(x2+xy+y2)+2(x−y)2f

5、orallrealsx,yResult15.x2+y2+z2−(xy+yz+zx)=(x−y)2+(x−z)(y−z)forallrealsx,y,zResult16.x2+y2+z2−(xy+yz+zx)=1[(2x−y−z)2+(2y−z−x)2+(2z−x−y)2]forallrealsx,y,z6Result17.(xy+yz+zx)(x+y+z)=(x2y+y2z+z2x)+(xy2+yz2+zx2)+3xyzforallrealsx,y,zResult18.222222(x+y)(y+z)(

6、z+x)=(xy+yz+zx)+(xy+yz+zx)+2xyzforallrealsx,y,zResult19.(xy+yz+zx)(x+y+z)=(x+y)(y+z)(z+x)+xyzforallrealsx,y,zResult20.3(x+y)(y+z)(z+x)=(x+y+z)3−(x3+y3+z3)forallrealsx,y,zResult21.(x−y)(y−z)(z−x)=(xy2+yz2+zx2)−(x2y+y2z+z2x)forallrealsx,y,zResult22.(x−y)(y

7、−z)(z−x)(x+y+z)=(xy3+yz3+zx3)−(x3y+y3z+z3x)forallrealsx,y,zResult23.(x−y)(y−z)(z−x)(xy+yz+zx)=(x2y3+y2z3+z2x3)−(x3y2+y3z2+z3x2)forallrealsx,y,zResult24.x3+y3+z3−3xyz=1(x+y+z)[(x−y)2+(y−z)2+(z−x)2]forallrealsx,y,z2Result25a3+b3+c3−3abc=(a+b+c)(a−b)2+(a+b+

8、c)(a−c)(b−c)forallreala,b,cResult26(a+b)(b+c)(c+a)−8abc=2c(a−b)2+(a+b)(a−c)(b−c)forallreala,b,cResult27ab2+bc2+ca2−3abc=c(a−b)2+b(a−c)(b−c)forallreala,b,cResult28a4+b4+c4−a3b−b3c−c3a=(a2+ab+b2)(a−b)2+(b2+bc+c2)(a−c)(b−c)fo