Question

Chứng minh các đa thức sau vô nghiệm: 
G(x) = x² +2x +3 
A(x) = x² -x +1

Answer 1

\(G\left(x\right)=x^2+2x+3\)

          \(=x^2+x+x+1+2\)

          \(=x.\left(x+1\right)+\left(x+1\right)+2\)

          \(=\left(x+1\right).\left(x+1\right)+2\)

          \(=\left(x+1\right)^2+2\ge2\)

Vậy \(G\left(x\right)\) vô nghiệm .

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

         \(=x^2-\frac{1}{2}x-\frac{1}{2}x+\frac{1}{4}+\frac{3}{4}\)

         \(=x.\left(x-\frac{1}{2}\right)-\frac{1}{2}.\left(x-\frac{1}{2}\right)+\frac{3}{4}\)

         \(=\left(x-\frac{1}{2}\right).\left(x-\frac{1}{2}\right)+\frac{3}{4}\)

        \(=\left(x-\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\)

Vậy \(A\left(x\right)\) vô nghiệm