Question

Chứng tỏ rằng thương của phép chia sau luôn dương với mọi x

[(X^4+1)^5-2(x^4+1)^4+3 (x^4+1)^3] : (x^4+1)^3

Answer 1

Lời giải:

Xét:

\(\left(x^4+1\right)^5-2\left(x^4+1\right)^4+3\left(x^4+1\right)^3\)

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

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

Trở lại bài toán:

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

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

\(=x^8+2x^4+1-2x^4-2+3\)

\(=x^8+2>0\left(đpcm\right)\)