Find the polynomial whose sum and product of the zeros are $\frac{2}{3}$ and $\frac{5}{3}$ respectively.
Given :
The sum and the product of zeros of the polynomial are $\frac{2}{3}$ and $\frac{5}{3}$.
To do :
We have to find the polynomial.
Solution :
Let $\alpha$ and $\beta$ are the roots of the required polynomial.
So, $\alpha + \beta = \frac{2}{3}$
$\alpha \times \beta = \frac{5}{3}$
If $\alpha$ and $\beta$ are the roots of the polynomial, then the polynomial is,
$x^2 -(\alpha + \beta)x + ( \alpha \times \beta) = 0$
$x^2 - \frac{2}{3}x + \frac{5}{3} = 0$
Multiply 3 on both sides, we get,
$3x^2 - 2x + 5 = 0$
Therefore, the required polynomial is $3x^2 - 2x + 5= 0$.
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