Answer:
[tex]f(x) = 2x^3 -5x^2+2x+1\\\\by \ trail \ and \ error, \\\\f(1) = 2(1^3) -5(1^2) +2(1) +1 = 2 - 5 + 2 + 1 = 0\\\\Therefore, 1 \ is \ a \ root \ of \ f(x).\\\\=> (x -1) \ is \ a \ factor \ of \ f(x).[/tex]
Now using long division,
[tex]\mathrm{Factor}\:2x^3-5x^2+2x+1:\quad \left(x-1\right)\left(2x^2-3x-1\right)[/tex]
[tex]\quad \left(x-1\right)\left(2x^2-3x-1\right)\\\\(x-1)(x-\frac{3+\sqrt{17} }{4})(x+\frac{3-\sqrt{17} }{4} )[/tex]
Therefore , the roots are
[tex]1 , \frac{3+\sqrt{17} }{4} , \frac{3-\sqrt{17} }{4}[/tex]
