Question

Phân tích các đa thức sau thành nhân tử

a) ( x + y + z )³ - x³ - y³ - z³

b) x⁴ + 2010x² + 2009x + 2010

Answer 1

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

\(=\left[\left(x+y\right)^3+z^3\right]-a^3-b^3-c^3\)

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

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

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

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

\(=3\left(x+y\right)\left(y+z\right)\left(z+x\right)\)

b.\(x^4+2010x^2+2009x+2010\)

\(=\left(x^4-x\right)+\left(2010x^2+2010x+2010\right)\)

=\(x\left(x-1\right)\left(x^2+x+1\right)+2010\left(x^2+x+1\right)\)

=\(\left(x^2+x+1\right)\left(x^2-x+2010\right)\)