Find a quadratic polynomial with the given numbers as the sum and product of zeroes respectively: $-\frac{1}{4},\ \frac{1}{4}$
Given: The sum and product of zeroes respectively are : $-\frac{1}{4},\ \frac{1}{4}$
To do: To find the quadratic polynomial.
Solution:
Let $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial.
As given, Sum of the zeroes$=\alpha +\beta=-\frac{1}{4}$
Product of the zeroes$=\alpha \beta =\frac{1}{4}$
$x^{2}-( \alpha+\beta)+\alpha\beta =0$
$x^{2}-( -\frac{1}{4})x+\frac{1}{4}=0$
$4x^{2}+x+1=0$
Thus, he required quadratic polynomial is $4x^2+x+1$
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