Đáp án:
`a, x^4 + 64 = x^4 + 16x^2 + 64 - 16x^2`
`= [(x^2)^2 + 2.x^2 .8 + 8^2] - 16x^2`
`=(x^2 + 8)^2 - (4x)^2`
`= (x^2 + 4x + 8)(x^2 - 4x + 8)`
`b, x^4 + 4y^4 = x^4 + 4x^2 y^2 + 4y^4 - 4x^2 y^2`
`= (x^4 + 4x^2 y^2 + 4y^4) - 4x^2 y^2`
`= [(x^2)^2 + 2.x^2 .2y^2 + (2y)^2] - 4x^2 y^2`
`= (x^2 + 2y^2)^2 - 4x^2 y^2`
`= (x^2 + 2y^2)^2 - (2xy)^2`
`= (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)`
`c, x^5 + x^4 + 1`
`= x^5 + x^4 + x^3 - x^3 - x^2 - x + x^2 + x + 1`
`= (x^5 + x^4 + x^3) - (x^3 + x^2 + x) + (x^2 + x + 1)`
`= x^3 (x^2 + x + 1) - x(x^2 + x + 1) + (x^2 + x + 1)`
`= (x^2 + x + 1)(x^3 - x + 1)`
`d, x^8 + x^4 + 1`
`= x^8 + x^4 + x^4 + 1 - x^4`
`=[ (x^4)^2 + 2x^4 .1 + 1^2 ]- x^4`
`= (x^4 + 1)^2 - x^4`
`= (x^4 + 1)^2 - (x^2)^2`
`= (x^4 + x^2 + 1)(x^4 - x^2 + 1)`
`e, x^7 + x^2 + 1`
`= x^7 + x^6 + x^5 - x^6 - x^5 - x^4 + x^4 + x^3 + x^2 - x^3 - x^2 - x + x^2 + x + 1`
`= (x^7 + x^6 + x^5) - (x^6 + x^5 + x^4) + (x^4 + x^3 + x^2) - (x^3 + x^2 + x) + (x^2 + x + 1)`
`= x^5 (x^2 + x + 1) - x^4 (x^2 + x + 1) + x^2 (x^2 + x + 1) - x(x^2 + x + 1) + (x^2 + x + 1)`
`= (x^2 + x + 1)(x^5 - x^4 + x^2 - x + 1)`
`f, x^10 + x^5 + 1`
`= x^10 - x + x^5 - x^2 + x^2 + x + 1`
`= (x^10 - x) + (x^5 - x^2) + (x^2 + x + 1)`
`= x(x^9 - 1) + x^2 (x^3 - 1) + (x^2 + x + 1)`
`= x[(x^3)^3 - 1^3] + x^2 (x - 1)(x^2 + x + 1) + (x^2 + x + 1)`
`= x(x^3 - 1)(x^6 + x^3 + 1) + x^2 (x - 1)(x^2 + x + 1) + (x^2 + x + 1)`
`= x(x - 1)(x^2 + x + 1)(x^6 + x^3 + 1) + x^2 (x - 1)(x^2 + x + 1) + (x^2 + x + 1)`
`= (x^2 + x + 1)[x(x - 1)(x^6 + x^3 + 1) + x^2 (x - 1) + 1]`
`= (x^2 + x + 1)[(x^2 - x)(x^6 + x^3 + 1) + x^3 - x^2 + 1]`
`= (x^2 + x + 1)(x^8 + x^5 + x^2 - x^7 - x^4 - x + x^3 - x^2 + 1)`
`= (x^2 + x + 1)(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)`
`g, x^3 + y^3 + z^3 - 3xyz`
`= (x + y)^3 - 3xy(x + y) + z^3 - 3xyz`
`= [(x + y)^3 + z^3] - [3xy(x + y) + 3xyz]`
`= (x + y + z)^3 - 3(x + y)z(x + y + z) - 3xy(x + y + z)`
`= (x + y + z)[(x + y + z)^2 - 3(x + y)z - 3xy)`
`= (x + y + z)(x^2 + y^2 + z^2 + 2xy + 2yz + 2xz - 3xz - 3yz - 3xy)`
`= (x + y + z)(x^2 + y^2 + z^2 - xy - yz - xz)`
* Áp dụng các hằng đẳng thức:
`A^3 + B^3 = (A + B)^3 - 3AB(A + B)`
`(A + B + C)^2 = A^2 + B^2 + C^2 + 2AB + 2BC + 2AC`
