Question

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively,
$\sqrt{2},\frac{1}{3}$

Answer 1

Let the polynomial be

$p\left(x\right)=a{x}^2+bx+c$

Sum of zeroes $=\sqrt{2}$

$-\frac{b}{a}=\sqrt{2}$

Asuuming $a=1$

$-\frac{b}{1}=\sqrt{2}$

$b=-\sqrt{2}$

Product of zeroes $=-1$

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

Assuming $a=1$

$c=\frac{1}{3}$

Now $a=1,b=-\sqrt{2}$ and $c=\frac{1}{3}$

Hence, the required quadratic polynomial $=a{x}^2+bx+c$

$=x^2-\sqrt{2}x+\frac{1}{3}$

$=3(x^2-\sqrt{2}x+\frac{1}{3})$

$=3x^2-3\sqrt{2}x+1$