Find a quadratic polynomial with the given numbers as the sum and product of zeroes respectively. $0,\ \sqrt{5}$
Given: The sum and product of zeroes of quadratic polynomial is: $0,\ \sqrt{5}$ respectively.
To do: To find the quadratic polynomial.
Solution:
Let $\alpha$ and $\beta$ be the zeroes of the quadratic polynomial.
Sum of the zeroes of quadratic polynomial$=\alpha+\beta=0$
Product of the quadratic polynomial$=\alpha\beta=\sqrt{5}$
The quadratic polynomial is:
$x^{2}-(\alpha +\beta )+\alpha\beta=0$
$\Rightarrow x^{2}-0x+\sqrt{5}=0$
$\Rightarrow x^{2}+\sqrt{5}=0$
The required quadratic polynomial is $x^2+\sqrt{5}$.
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