Phân tích đa thức thành nhân tử a.[x+y+z]^3-x^3-y^3-z^3 b.x^4+2010.x^2+2009x+2010 26/07/2021 Bởi Adalynn Phân tích đa thức thành nhân tử a.[x+y+z]^3-x^3-y^3-z^3 b.x^4+2010.x^2+2009x+2010
a. $(x+y+z)^3-x^3-y^3-z^3$ $=[(x+y)+z]^3-(x^3+y^3)-z^3$ $=(x+y)^3+3(x+y)^2z+3(x+y)z^2+z^3-(x+y)(x^2-xy+y^2)-z^3$ $=(x+y)[(x+y)^2+3(x+y)z+3z^2-(x^2-xy+y^2)]$ $=(x+y)(x^2+2xy+y^2+3xz+3yz+3z^2-x^2+xy-y^2)$ $=(x+y)(3xy+3xz+3yz+3z^2)$ $=3(x+y)[(xy+xz)+(yz+z^2)]$ $=3(x+y)(y+z)(x+z)$ b. $x^4+2010x^2+2009x+2010$ $=(x^4+x^3+x^2)-(x^3+x^2+x)+(2010x^2+2010x+2010)$$=(x^2+x+1)(x^2-x+2010)$. Bình luận
Đáp án: a. ` 3(y + z)(x+y)(x+z)` b. ` (x^2 + x + 1)(x^2 – x + 2010)` Giải thích các bước giải: a. `(x + y + z)^3 – x^3 – y^3 – z^3` `= [(x + y + z)^3 – x^3) – (y^3 + z^3)` `= (x + y + z – x)^3 + 3(x + y + z)x(x + y + z – x) – (y + z)^3 + 3yz(y + z)` `= (y+z)^3 + 3(x+y+z)x(y+z) – (y + z)^3 + 3yz(y + z)` `= (y + z)[(y + z)^2 + 3(x+y+z)x – (y + z)^2 + 3yz]` `= (y + z)(3x^2 + 3xy + 3xz + 3yz)` `= (y + z)[(3x^2 + 3xy)+(3xz + 3yz)]` `= (y + z)[3x(x + y) + 3z(x + y)]` `= (y + z)(x + y)(3x + 3z)` `= 3(y + z)(x+y)(x+z)` b. `x^4 + 2010x^2 + 2009x + 2010` `= x^4 + 2010x^2 + 2010x – x + 2010` `= (x^4 -x) + (2010x^2 + 2010x + 2010)` `= x(x^3 – 1) + 2010(x^2 + x + 1)` `= x(x – 1)(x^2 + x + 1) + 2010(x^2 + x + 1)` `= (x^2 – x)(x^2 + x + 1) + 2010(x^2 + x + 1)` `= (x^2 + x + 1)(x^2 – x + 2010)` Bình luận
a. $(x+y+z)^3-x^3-y^3-z^3$
$=[(x+y)+z]^3-(x^3+y^3)-z^3$
$=(x+y)^3+3(x+y)^2z+3(x+y)z^2+z^3-(x+y)(x^2-xy+y^2)-z^3$
$=(x+y)[(x+y)^2+3(x+y)z+3z^2-(x^2-xy+y^2)]$
$=(x+y)(x^2+2xy+y^2+3xz+3yz+3z^2-x^2+xy-y^2)$
$=(x+y)(3xy+3xz+3yz+3z^2)$
$=3(x+y)[(xy+xz)+(yz+z^2)]$
$=3(x+y)(y+z)(x+z)$
b. $x^4+2010x^2+2009x+2010$
$=(x^4+x^3+x^2)-(x^3+x^2+x)+(2010x^2+2010x+2010)$
$=(x^2+x+1)(x^2-x+2010)$.
Đáp án:
a. ` 3(y + z)(x+y)(x+z)`
b. ` (x^2 + x + 1)(x^2 – x + 2010)`
Giải thích các bước giải:
a. `(x + y + z)^3 – x^3 – y^3 – z^3`
`= [(x + y + z)^3 – x^3) – (y^3 + z^3)`
`= (x + y + z – x)^3 + 3(x + y + z)x(x + y + z – x) – (y + z)^3 + 3yz(y + z)`
`= (y+z)^3 + 3(x+y+z)x(y+z) – (y + z)^3 + 3yz(y + z)`
`= (y + z)[(y + z)^2 + 3(x+y+z)x – (y + z)^2 + 3yz]`
`= (y + z)(3x^2 + 3xy + 3xz + 3yz)`
`= (y + z)[(3x^2 + 3xy)+(3xz + 3yz)]`
`= (y + z)[3x(x + y) + 3z(x + y)]`
`= (y + z)(x + y)(3x + 3z)`
`= 3(y + z)(x+y)(x+z)`
b. `x^4 + 2010x^2 + 2009x + 2010`
`= x^4 + 2010x^2 + 2010x – x + 2010`
`= (x^4 -x) + (2010x^2 + 2010x + 2010)`
`= x(x^3 – 1) + 2010(x^2 + x + 1)`
`= x(x – 1)(x^2 + x + 1) + 2010(x^2 + x + 1)`
`= (x^2 – x)(x^2 + x + 1) + 2010(x^2 + x + 1)`
`= (x^2 + x + 1)(x^2 – x + 2010)`