Chứng minh: x^4 + y^4 + (x+y)^4 = 2(x ²+xy+y ²) ² 10/07/2021 Bởi Melody Chứng minh: x^4 + y^4 + (x+y)^4 = 2(x ²+xy+y ²) ²
`VT=x^4+y^4+(x+y)^4` `=x^4+y^4+[(x+y)²]²` `=x^4+y^4+(x²+2xy+y²)²` `=x^4+y^4+[(x²+y²)+2xy]²` `=x^4+y^4+(x²+y²)²+2(x²+y²).2xy+(2xy)²` `=x^4+y^4+(x²)²+2.x².y²+(y²)²+4xy(x²+y²)+4x²y²` `=x^4+y^4+x^4+2x²y²+y^4+4x³y+4xy³+4x²y²` `=2x^4+2y^4+6x²y²+4x³y+4xy³` `=2(x^4+y^4+3x²y²+2x³y+2xy³)` `=2(x^4+y^4+2x²y²+x²y²+2x³y+2xy³)` `=2[(x^4+2x²y²+y^4)+(2x³y+2xy³)+x²y²]` `=2{[(x²)²+2.x².y²+(y²)²]+2(x²+y²).xy+(xy)²}` `=2[(x²+y²)²+2(x²+y²).xy+(xy)²]` `=2(x²+y²+xy)²` Vế trái bằng vế phải đẳng thức được chứng minh Bình luận
`x^4 + y^4 + (x+y)^4` `=x^4+y^4+(x^2+2xy+y^2)^2` `=x^4+y^4+x^4+y^4+2x^2y^2+4xy^3+4yx^3+4x^2y^2` `=2(x^4+y^4+3x^2y^2+2xy^3+2yx^3)` `=2(x^2(x^2+xy+y^2)+xy(x^2+xy+y^2)+y^2(x^2+xy+y^2))` `=2(x^2+xy+y^2)^2` Bình luận
`VT=x^4+y^4+(x+y)^4`
`=x^4+y^4+[(x+y)²]²`
`=x^4+y^4+(x²+2xy+y²)²`
`=x^4+y^4+[(x²+y²)+2xy]²`
`=x^4+y^4+(x²+y²)²+2(x²+y²).2xy+(2xy)²`
`=x^4+y^4+(x²)²+2.x².y²+(y²)²+4xy(x²+y²)+4x²y²`
`=x^4+y^4+x^4+2x²y²+y^4+4x³y+4xy³+4x²y²`
`=2x^4+2y^4+6x²y²+4x³y+4xy³`
`=2(x^4+y^4+3x²y²+2x³y+2xy³)`
`=2(x^4+y^4+2x²y²+x²y²+2x³y+2xy³)`
`=2[(x^4+2x²y²+y^4)+(2x³y+2xy³)+x²y²]`
`=2{[(x²)²+2.x².y²+(y²)²]+2(x²+y²).xy+(xy)²}`
`=2[(x²+y²)²+2(x²+y²).xy+(xy)²]`
`=2(x²+y²+xy)²`
Vế trái bằng vế phải đẳng thức được chứng minh
`x^4 + y^4 + (x+y)^4`
`=x^4+y^4+(x^2+2xy+y^2)^2`
`=x^4+y^4+x^4+y^4+2x^2y^2+4xy^3+4yx^3+4x^2y^2`
`=2(x^4+y^4+3x^2y^2+2xy^3+2yx^3)`
`=2(x^2(x^2+xy+y^2)+xy(x^2+xy+y^2)+y^2(x^2+xy+y^2))`
`=2(x^2+xy+y^2)^2`