YLL討論網

設實數a，b，x，y滿足
a+b=8
ax+by=9
a(x^2)+b(y^2)=57
a(x^3)+b(y^3)=111
求a(x^4)+b(y^4)=?

1.先將ax+by乘上x+y

(ax+by)(x+y)=ax^2+axy+bxy+by^2

=> 9(x+y)=ax^2+by^2+axy+bxy
=> 9(x+y)=57+(a+b)xy
=> 9(x+y)=57+8xy --------------------------(1)

2.再將ax^2+by^2乘上x+y

(ax^2+by^2)(x+y)=ax^3+ax^2y+bxy^2+by^3

=> 57(x+y)=(ax^3+by^3)+(ax+by)xy
=> 57(x+y)=111+9xy------------------------(2)

3.
9(x+y)=57+8xy --------------------------(1)x9
57(x+y)=111+9xy------------------------(2)x8

81(x+y)=513+72xy------------(3)
456(x+y)=888+72xy-----------(4)

(4)-(3)

375(x+y)=375

(x+y)=1   代入

得到  xy= -6

4.最後將ax^3+by^3乘上x+y

(ax^3+by^3)(x+y)=ax^4+ax^3y+bxy^3+by^4

=>  111．1= (ax^4+by^4)+(ax^2+by^2)xy
=>  111 = (ax^4+by^4)+ 57．(- 6)
=>  111+342 =ax^4+by^4 = 453
=>  ax^4+by^4 = 453