Question

Find a quadratic polynomial for the given numbers as the sum and product of its zeroes respectively.

$\sqrt{2},\frac{1}{3}$

Answer 1

$\sqrt{2},\frac{1}{3}$

Let the polynomial be $a{x}^2+bx+c$, and its zeroes be $\alpha and\beta$.

$\alpha +\beta =\sqrt{2}$.

$\alpha \beta =\frac{1}{3}=\frac{c}{a}$

$\therefore \alpha +\beta$ = $\frac{3\sqrt{2}}{3}=\frac{-b}{a}$

$\therefore$ a = 3, b = -3$\sqrt{2}$, c = 1

$\therefore$ Quadratic polynomial is $3x^2-3\sqrt{2}x+1$.