Question

[nguyen trung tai]

cmr bieu thuc nay luon am

P=x+x^2+x^3-x^4-x^5+x^6-x^7-1

Answer 1

[Nguyễn Lê Phước Thịnh]

\(P=x^6\left(1-x\right)+x^3\left(x^2-1\right)-x^2\left(x^2-1\right)+x-1\)

\(=\left(x-1\right)\left(-x^6+x^3\left(x+1\right)-x^2\left(x+1\right)+1\right)\)

\(=\left(x-1\right)\left[x^2\left(x+1\right)\left(x-1\right)-\left(x^6-1\right)\right]\)

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

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

\(=\left(x-1\right)^2\cdot\left(x+1\right)\left(-x^4-1\right)< 0\)