Obtain all zeros of the polynomial 2x3-4x-x2+2,if two of its zeros are

1.	Obtain all zeros of the polynomial 2x3-4x-x2+2,if two of its zeros are √2 and -√2
Answer» The given polynomial is:f(x) = 2x3\xa0- x2 - 4x + 2.It is given that the two zeroes of the above polynomial are\xa0{tex}\\sqrt{2}{/tex}\xa0and -{tex}\\sqrt{2}{/tex}Therefore, (x - {tex}\\sqrt{2}{/tex})(x + {tex}\\sqrt{2}{/tex}) = (x2 -\xa02) is a factor of f(x).Now we divide (x) = 2x3\xa0- x2 - 4x + 2 by (x2- 2), we obtainWhere quotient =\xa0(2x - 1){tex}\\therefore{/tex} f(x) = 0 {tex}\\Rightarrow{/tex}\xa0(x2\xa0- 2)(2x - 1) = 0{tex}\\Rightarrow{/tex}\xa0(x - {tex}\\sqrt{2}{/tex}) (x + {tex}\\sqrt{2}{/tex})(2x - 1) = 0{tex}\\Rightarrow{/tex}\xa0(x - {tex}\\sqrt{2}{/tex}) = 0 or (x + {tex}\\sqrt{2}{/tex}) = 0 or (2 x -1) = 0{tex}\\Rightarrow{/tex}\xa0x = {tex}\\sqrt{2}{/tex}\xa0or x = - {tex}\\sqrt{2}{/tex}\xa0or x = {tex}\\frac{1}{2}{/tex}.Hence, all zeros of f(x) are {tex}\\sqrt{2}{/tex}, -{tex}\\sqrt{2}{/tex}\xa0and\xa0{tex}\\frac{1}{2}{/tex}.

Obtain all zeros of the polynomial 2x3-4x-x2+2,if two of its zeros are √2 and -√2