已知x+y=2,x³+y³=14求x四次方+y四次方,x七次方+y七次方

已知x+y=2,x³+y³=14求x四次方+y四次方,x七次方+y七次方
已知x+y=2,x³+y³=14求x四次方+y四次方,x七次方+y七次方

已知x+y=2,x³+y³=14求x四次方+y四次方,x七次方+y七次方
x+y=2
(x+y)^2=x^2+y^2+2xy=4(1)
x³+y³=(x+y)(x^2-xy+y^2)=2(x^2-xy+y^2)=14
x^2+y^2-xy=7(2)
(1)-(2)得：xy=-1
即：x^2+y^2=6
x^4+y^4=(x^2+y^2)^2-2x^2y^2=36-2=34
x^7+y^7=(x^3+y^3)(x^4+y^4)-(x+y)x^3y^3
=14*34-2*(-1)^3
=478
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x+y=2，①
x³+y³=14②
由②得
(x+y)(x²-xy+y²)=14
x²-xy+y²=7③
①的平方-③得
xy=-1④
联立①④得
x或y=1±√2
x四次方+y四次方=(1+√2)四次方+(1-√2)四次方
=[(1+√2)²...

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x+y=2，①
x³+y³=14②
由②得
(x+y)(x²-xy+y²)=14
x²-xy+y²=7③
①的平方-③得
xy=-1④
联立①④得
x或y=1±√2
x四次方+y四次方=(1+√2)四次方+(1-√2)四次方
=[(1+√2)²]² +[(1-√2)²]²
=(3+2√2)²+(3-2√2)²
=34
x七次方+y七次方=(1+√2)^4*(1+√2)^3 +(1-√2)^4(1-√2)^3
=(17+12√2)(3+2√2)(1+√2)+(17-12√2)(3-2√2)(1-√2)
=(17+12√2)(7+5√2)+(17-12√2)(7-5√2)
=119+120+169√2+119+120-169√2
=478

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x³+y³=(x+y)(x^2-xy+y^2)=14
x^2-xy+y^2=7
x^2+2xy+y^2=4 xy=-1 x^2+y^2=6
x^4+y^4=(x^2+y^2)^2-2x^2y^2=36-2=34
x^7+y^7=(x^3+y^3)(x^4+y^4)-x^3y^4-x^4y^3=14*34-x^3y^3(x+y)=478