Question

Chứng minh rằng: x³(x² - 7) - 36x chia hết cho 7 với mọi số nguyên x.

Answer 1

Mạn phép sửa đề \(x^3\left(x^2-7\right)^2-36x\)

\(x\left(x^2\left(x^2-7\right)^2-36\right)=x\left(\left(x^2-7x\right)^2-6^2\right)=x\left(x^3-7x+6\right)\left(x^3-7x-6\right)=x\left(\left(x^3+1\right)-\left(7x+7\right)\right)\left(\left(x^3-x\right)-\left(6x-6\right)\right)=x\left(\left(x+1\right)\left(x^2-x+1\right)-7\left(x+1\right)\right)\left(x\left(x+1\right)\left(x-1\right)-6\left(x-1\right)\right)=x\left(x+1\right)\left(x^2-x-6\right)\left(x-1\right)\left(x^2+x-6\right)=x\left(x+1\right)\left(x-3\right)\left(x+2\right)\left(x-1\right)\left(x-2\right)\left(x+3\right)\)chia ht 7