已知x＋y＝3,x²＋y²－xy＝4,求x⁴＋y⁴＋x³y＋xy³的值

已知x＋y＝3,x²＋y²－xy＝4,求x⁴＋y⁴＋x³y＋xy³的值
已知x＋y＝3,x²＋y²－xy＝4,求x⁴＋y⁴＋x³y＋xy³的值

已知x＋y＝3,x²＋y²－xy＝4,求x⁴＋y⁴＋x³y＋xy³的值
x⁴＋y⁴＋x³y＋xy³
=(x⁴+x³y)+(xy³+y⁴)
=x³(x+y)+y³(x+y)
=(x+y)(x³+y³)
=(x+y)(x+y)(x²+y²-xy)
=(x+y)²(x²+y²-xy)
=3²×4
=36

x⁴+y⁴+x³y+xy³
=(x³+y³)(x+y)
=[(x+y)(x²-xy+y²)](x+y)
=(x²-xy+y²)(x+y)²
=4×3²
=36
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