Question

Phân tích đa thức thành nhân tử

a) \(\left(x+y+z\right)^3-x^3-y^3-z^3\)

b) \(\left(b-c\right)^3+\left(c-a\right)^3+\left(a-b\right)^3\)

c) \(x^3+y^3+z^3-3xyz\)

d) \(\left(x^2+y^2\right)^3+\left(z^2-x^2\right)^3-\left(y^2+z^2\right)^3\)

e) \(x^3-5x^2y-14xy^2\)

Answer 1

a)

\((x+y+z)^3-x^3-y^3-z^3\)

\(=(x+y+z-x)[(x+y+z)^2+x(x+y+z)+x^2]-(y^3+z^3)\)

\(=(y+z)(x^2+y^2+z^2+2xy+2yz+2xz+x^2+xy+xz+x^2)-(y+z)(y^2-yz+z^2)\)

\(=(y+z)(x^2+y^2+z^2+2xy+2yz+2xz+x^2+xy+xz+x^2-y^2+yz-z^2)\)

\(=(y+z)(3x^2+3xy+3yz+3xz)\)

\(=3(y+z)(x^2+xy+yz+xz)\)

\(=3(y+z)[x(x+y)+z(y+x)]=3(y+z)(x+z)(y+x)\)

Answer 2

b)

\((b-c)^3+(c-a)^3+(a-b)^3\)

\(=(b-c)^3-[(b-c)+(a-b)]^3+(a-b)^3\)

\(=(b-c)^3-[(b-c)^3+3(b-c)^2(a-b)+3(b-c)(a-b)^2+(a-b)^3]+(a-b)^3\)

\(=-3(b-c)^2(a-b)-3(b-c)(a-b)^2\)

\(-3(b-c)(a-b)[(b-c)+(a-b)]=-3(b-c)(a-b)(a-c)\)

\(=3(a-b)(b-c)(c-a)\)

e)

\(x^3-5x^2y-14xy^2\)

\(=x(x^2-5xy-14y^2)\)

\(=x[x^2+2xy-7xy-14y^2]\)

\(=x[x(x+2y)-7y(x+2y)]\)

\(=x(x-7y)(x+2y)\)

Answer 3

c)

\(x^3+y^3+z^3-3xyz\)

\(=(x+y)^3-3xy(x+y)+z^3-3xyz\)

\(=[(x+y)^3+z^3]-3xy(x+y+z)\)

\(=(x+y+z)[(x+y)^2-z(x+y)+z^2]-3xy(x+y+z)\)

\(=(x+y+z)(x^2+2xy+y^2-xz-zy+z^2-3xy)\)

\(=(x+y+z)(x^2+y^2+z^2-xy-yz-xz)\)

Answer 4

d)

\((x^2+y^2)^3+(z^2-x^2)^3-(y^2+z^2)^3\)

\(=[(x^2+y^2)^3-(y^2+z^2)^3]+(z^2-x^2)^3\)

\(=(x^2-z^2)[(x^2+y^2)^2+(x^2+y^2)(y^2+z^2)+(y^2+z^2)^2]-(x^2-z^2)^3\)

\(=(x^2-z^2)[(x^2+y^2)^2+(x^2+y^2)(y^2+z^2)+(y^2+z^2)^2-(x^2-z^2)^2]\)

\(=(x^2-z^2)(x^4+2x^2y^2+y^4+x^2y^2+x^2z^2+y^4+y^2z^2+y^4+z^4+2y^2z^2-x^4-z^4+2x^2z^2)\)

\(=(x^2-z^2)(3y^4+3x^2y^2+3x^2z^2+3y^2z^2)\)

\(=3(x-z)(x+z)(y^4+x^2y^2+x^2z^2+y^2z^2)\)

\(=3(x-z)(x+z)[y^2(y^2+x^2)+z^2(x^2+y^2)]\)

\(=3(x-z)(x+z)(y^2+x^2)(y^2+z^2)\)