Question

Giải bất phương trình

3x\(^3\) - 5x\(^2\) - x - 2>0

Answer 1

=>3x^3-6x^2+x^2-2x+x-2>0

=>(x-2)(3x^2+x+1)>0

=>x-2>0

=>x>2

Answer 2

 \(3x^3-5x^2-x-2>0\)

\(\Leftrightarrow3x^3-6x^2+x^2-2x+x-2>0\)

\(\Leftrightarrow3x^2\left(x-2\right)+x\left(x-2\right)+\left(x-2\right)>0\)

\(\Leftrightarrow\left(x-2\right)\left(3x^2+x+1\right)>0\)

Mặt khác: \(3x^2+x+1=2x^2+\left(x^2+x+1\right)\)

Ta lại có: \(x^2+x+1=x^2+2x\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{3}{4}=\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}\)

\(\Rightarrow3x^2+x+1>0\)

\(\Rightarrow x-2>0\)

\(\Leftrightarrow x>2\)

Vậy bpt có nghiệm là \(x>2\)