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

If $\alpha$ and $\beta$ are the zeroes of a quadratic polynmial, then the polynomial can be written as follows-

$p\left(x\right)=x^2-(\alpha +\beta )x+\left(\alpha \beta \right)$

Whereas

$\alpha +\beta =$ Sum of roots

$\alpha \beta =$ Product of roots

Now according to the question,

Sum of roots $=\alpha +\beta =\sqrt{2}$

Product of roots $=\alpha \beta =\frac{1}{3}$

Therefore the quadratic polynomial will be-

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

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

Hence the quadratic polynomial will be $x^2+\sqrt{2}x+\frac{1}{3}$.