find the quadratic polynomial ,the sum …

Let the polynomial is f(x) and zeros are α and β

then f(x)=x²-(α+β)x+ αβ

Given {tex}\alpha + \beta = \frac { 5 } { 2 } , \alpha \beta = 1{/tex}
 {tex}x ^ { 2 } - ( \alpha + \beta ) x + \alpha \beta{/tex}

f(x)={tex}= x ^ { 2 } - \frac { 5 } { 2 } x + 1 = \frac { 1 } { 2 } \left( 2 x ^ { 2 } - 5x + 2 \right){/tex}
The polynomial whose zero are {tex}\alpha , \beta \text { is } 2 x ^ { 2 } - 5 x + 2{/tex}
Further, {tex}f ( x ) = \frac { 1 } { 2 } \left( 2 x ^ { 2 } - 5 x + 2 \right) = \frac { 1 } { 2 } \left( 2 x ^ { 2 } - 4 x - x + 2 \right){/tex}
{tex}= \frac { 1 } { 2 } [ 2 x ( x - 2 ) - ( x - 2 ) ]{/tex}
{tex}= \frac { 1 } { 2 } ( x - 2 ) ( 2 x - 1 ){/tex}
f(x) = 0 {tex}\Rightarrow \frac { 1 } { 2 } ( x - 2 ) ( 2 x - 1 ) = 0{/tex}
{tex}\therefore{/tex} for that x - 2 = 0 or 2x - 1 = 0
i.e., Either x = 2 or {tex}x = \frac { 1 } { 2 }{/tex}
{tex}\therefore{/tex} Zeros of polynomial are 2 and {tex} \frac { 1 } { 2 }{/tex}.