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13) x²-x-y²-y
= x²-y²-x-y
= (x²-y²)-(x+y)
= (x-y)(x+y)-(x+y)
= (x-y-1)(x+y)
14) x²-2xy+y²-z²
= (x-y)²-z²
= (x-y-z)(x-y+z)
15) a³-a²x-ay+xy
= a²(a-x)-y(a-x)
= (a²-y)(a-x)
16) x²-2xy-4z²+y²
= (x²-2xy+y²)-4z²
= (x-y)²-(2z)²
= (x-y-2z)(x-y+2z)
17) 4x²-y²+4x+1
= (4x²+4x+1)-y²
= (2x+1)²-y²
= (2x+1-y)(2x+1+y)
18) x³-x+y³-y
= (x³+y³)-(x+y)
= (x+y)(x²-xy+y²)-(x+y)
= (x+y)(x²-xy+y²-1)
19) $x^{4}$ +2x³+x²
= x²(x²+2x+1)
= x²(x+1)²
20) 5x²-10xy+5y²-20z²
= 5(x²-2xy+y²-4z²)
= 5[(x+y)²-(2z)²]
= 5(x+y-2z)(x+y+2z)
21) x³-x+3x²y+3xy²+y³-y
= (x³+3x²y+3xy²+y³)-(x+y)
= (x+y)³-(x+y)
= (x+y) (x+y+1)(x+y-1)
Đáp án: $13)(x+y)(x-y-1)$
$14)(x-y-z)(x-y+z)$
$15)(a^2-y)(a-x)$
$16)(x-y-2z)(x-y+2z)$
$17)(2x+1-y)(2x+1+y)$
$18)(x+y)(x^2-xy+y^2-1)$
$19)x^2(x+1)^2$
$20)5(x-y-2z)(x-y+2z)$
$21)(x+y)(x+y-1)(x+y+1)$
Giải thích các bước giải:
$13)x^2-x-y^2-y$
$=(x^2-y^2)-(x+y)$
$=(x-y)(x+y)-(x+y)$
$=(x+y)(x-y-1)$
$14)x^2-2xy+y^2-z^2$
$=(x^2-2xy+y^2)-z^2$
$=(x-y)^2-z^2$
$=(x-y-z)(x-y+z)$
$15)a^3-a^2x-ay+xy$
$=(a^3-a^2x)-(ay-xy)$
$=a^2(a-x)-y(a-x)$
$=(a^2-y)(a-x)$
$16)x^2-2xy-4z^2+y^2$
$=(x^2-2xy+y^2)-4z^2$
$=(x-y)^2-(2z)^2$
$=(x-y-2z)(x-y+2z)$
$17)4x^2-y^2+4x+1$
$=(4x^2+4x+1)-y^2$
$=(2x+1)^2-y^2$
$=(2x+1-y)(2x+1+y)$
$18)x^3-x+y^3-y$
$=(x^3+y^3)-(x+y)$
$=(x+y)(x^2-xy+y^2)-(x+y)$
$=(x+y)(x^2-xy+y^2-1)$
$19)x^4+2x^3+x^2$
$=x^2(x^2+2x+1)$
$=x^2(x+1)^2$
$20)5x^2-10xy+5y^2-20z^2$
$=5(x^2-2xy+y^2-4z^2)$
$=5[(x^2-2xy+y^2)-4z^2]$
$=5[(x-y)^2-(2z)^2]$
$=5(x-y-2z)(x-y+2z)$
$21)x^3-x+3x^2y+3xy^2+y^3-y$
$=(x^3+3x^2y+3xy^2+y^3)-(x+y)$
$=(x+y)^3-(x+y)$
$=(x+y)[(x+y)^2-1]$
$=(x+y)(x+y-1)(x+y+1)$