Question

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively,
$\frac{1}{4},-1$

Answer 1

Let the polynomial be

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

Sum of zeroes $=\frac{1}{4}$

$-\frac{b}{a}=\frac{1}{4}$

Asuuming $a=1$

$-\frac{b}{1}=\frac{1}{4}$

$b=-\frac{1}{4}$

Product of zeroes $=-1$

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

Assuming $a=1$

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

$c=-1$

Now $a=1,b=-\frac{1}{4}$ and $c=-1$

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

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

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

$=4x^2-x-4$