Question

Phân tích đa thức thành nhân tử:

\(3\left(x^4+x^2+1\right)-\left(x^2+x+1\right)^2\)

Answer 1

Ta có :

\(3\left(x^4+x^2+1\right)-\left(x^2+x+1\right)^2\)

\(=3\left(x^4+x^3+x^2-x^3+1\right)-\left(x^2+x+1\right)^2\)

\(=3\left[\left(x^4+x^3+x^2\right)-\left(x^3-1\right)\right]-\left(x^2+x+1\right)^2\)

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

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

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

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

\(=\left(x^2+x+1\right)\left(2x^2+2-4x\right)\)

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

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