Find a quadratic polynomial of the …

Let the polynomial be ax² + bx + c
and its zeroes be {tex}\alpha{/tex} and {tex}\beta{/tex}
Then, {tex}\alpha + \beta = \frac { 1 } { 4 } = - \frac { b } { a } \text { and } \alpha \beta = - 1 = \frac { c } { a }{/tex}
If a = 4, then b = -1 and c = -4
So, one quadratic polynomial which files
the given conditions is 4x² - x - 4
Or
If {tex}\alpha{/tex} and {tex}\beta{/tex} zeroes of the polynomials then standard form quadratic polynomial is given by
{tex}x ^ { 2 } - ( \alpha + \beta ) x + \alpha \beta{/tex}
Let {tex}\alpha = \frac { 1 } { 4 } \text { and } \beta = - 1{/tex}
Now,
{tex}= x ^ { 2 } - ( \alpha + \beta ) x + \alpha \beta{/tex}
{tex}= x ^ { 2 } - \left( \frac { 1 } { 4 } \right) x + ( - 1 ){/tex}
{tex}= \frac { 1 } { 4 } \left( 4 x ^ { 2 } - x - 4 \right){/tex}
Required polynomial is 4x² - x - 4