Find a quadratic polynomial with the given numbers as the sum and product of zeroes respectively: $\sqrt{2},\ \frac{1}{3}$.
Given: The sum and the product of the zeroes of a quadratic polynomial are respectively: $\sqrt{2},\ \frac{1}{3}$.
To do: To find the quadratic polynomial.
Solution:
Let $\alpha$ and $\beta$ are the zeroes of the polynomial.
As given,
Sum of the zeroes$=\alpha +\beta=\sqrt{2}$
$\alpha\beta=\frac{1}{3}$
The quadratic polynomial is:
$x^{2}-( \alpha +\beta )+\alpha \beta =0$
$\Rightarrow x^{2}-\sqrt{2}x+\frac{1}{3}=0$
$\Rightarrow 3x^{2}-3\sqrt{2}x+1=0$
Thus, the required quadratic polynomial is $3x^{2}-3\sqrt{2}x+1$.
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