Question

Phân tích đa thức thành nhân tử :

a ) \(\left(a+b+c\right)^2+\left(a+b-c\right)^2-4c^2\)

b ) \(x^2-y^2+2x-4y-3\)

c ) \(xy\left(x-y\right)+yz\left(y-z\right)+zx\left(z-x\right)\)

d ) \(x^4+4a^4\)

e ) \(x^5+x+1\)

f ) \(x^4+2013x^2+2012x+2013\)

Answer 1

c) \(xy(x-y)+yz(y-z)+xz(z-x)\)

\(=xy(x-y)-yz[(x-y)+(z-x)]+zx(z-x)\)

\(=(xy-yz)(x-y)+(zx-yz)(z-x)\)

\(=y(x-z)(x-y)+z(x-y)(z-x)\)

\(=(x-y)(z-x)(z-y)\)

d) \(x^4+4a^4=(x^2)^2+(2a^2)^2\)

\(=(x^2)^2+(2a^2)^2+2x^2.2a^2-4x^2a^2\)

\(=(x^2+2a^2)^2-(2xa)^2\)

\(=(x^2+2a^2-2ax)(x^2+2a^2+2ax)\)

Answer 2

e)

\(x^5+x+1=x^5-x^2+x^2+x+1\)

\(=x^2(x^3-1)+x^2+x+1\)

\(=x^2(x-1)(x^2+x+1)+(x^2+x+1)\)

\(=(x^2+x+1)[x^2(x-1)+1]=(x^2+x+1)(x^3-x^2+1)\)

f)

\(x^4+2013x^2+2012x+2013\)

\(=x^4-x+2013x^2+2013x+2013\)

\(=x(x^3-1)+2013(x^2+x+1)\)

\(=x(x-1)(x^2+x+1)+2013(x^2+x+1)\)

\(=(x^2+x+1)[x(x-1)+2013]=(x^2+x+1)(x^2-x+2013)\)

Answer 3

a)

\((a+b+c)^2+(a+b-c)^2-4c^2\)

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

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

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

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

\(=2(a+b+c)(a+b-c)\)

b) \(x^2-y^2+2x-4y-3\)

\(=(x^2+2x+1)-(y^2+4y+4)\)

\(=(x+1)^2-(y+2)^2\)

\(=[(x+1)-(y+2)][(x+1)+(y+2)]\)

\(=(x-y-1)(x+y+3)\)