Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively

Solution:

(i) Required polynomial =

$x^{2}-($ sum of zeros $) x+$ product of zeros

$=x^{2}-\frac{1}{4} x-1$

$=\frac{1}{4}\left(4 x^{2}-x-1\right)$

(ii) Required polynomial =

$x^{2}-($ sum of zeros $) x+$ product of zeros

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

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

(iii) Required polynomial =

$x^{2}-(\operatorname{sum}$ of zeros $) x+$ product of zeros

$=x^{2}-0 x+\sqrt{5}$

$=x^{2}+\sqrt{5}$

(iv) Required polynomial =

$x^{2}-(\operatorname{sum}$ of zeros $) x+$ product of zeros

$=x^{2}-1 x+1$

$=x^{2}-x+1$

(v) Required polynomial =

$x^{2}-(\operatorname{sum}$ of zeros $) x+$ product of zeros

$=x^{2}-\left(-\frac{1}{4}\right) x+\frac{1}{4}$

$=x^{2}+\frac{1}{4} x+\frac{1}{4}$

$=\frac{1}{4}\left(4 x^{2}+x+1\right)$

(vi) Required polynomial =

$x^{2}-($ sum of zeros $) x+$ product of zeros $=x^{2}-4 x+1$