Find all the zeros of the …

Given polynomial is p(x)  = 2x⁴ - 3x³ - 5x² + 9x - 3
{tex}\sqrt { 3 } {/tex} and {tex}- \sqrt { 3 }{/tex} are the zeros of polynomial.
 (x - {tex}\sqrt{3}{/tex})(x + {tex}\sqrt{3}{/tex}) = x² - 3 will divide the given polynomial completely
Dividing 2x⁴ - 3x³ - 5x² + 9x - 3 by x² - 3, we get

{tex}\therefore{/tex} Quotient q(x) = 2x² - 3x + 1
= 2x² - 2x - x + 1
= 2x(x - 1) - 1(x - 1)
q(x) = (x - 1) (2x - 1)
Other zeros of given polynomial are given by
q(x) = 0
 {tex}\Rightarrow{/tex} (x - 1) (2x - 1) = 0
{tex}\Rightarrow{/tex} x - 1= 0 or 2x - 1 = 0
{tex}\Rightarrow{/tex} x = 1 or 2x = -1
{tex}\Rightarrow{/tex} x = 1 or x = {tex}\frac{1}{2}{/tex}
 {tex}\therefore{/tex} x = 1, {tex}\frac{1}{2}{/tex}
Hence, zeros of given polynomial are {tex}\sqrt { 3 } , - \sqrt { 3 }{/tex}, 1 , {tex}\frac{1}{2}{/tex}.