Question

Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
$4,1$

Answer 1

Let a Quadratic polynomial be $a{x}^2+bx+c,a\ne 0$

$\Rightarrow$ Sum of zeros $=-\frac{b}{a}$

and Product of zeros $=\frac{c}{a}$

It is given that, Sum of zeros $=4$, Product of zeros $=1$.

$\Rightarrow -\frac{b}{a}=4=\frac{4}{1}$ ---(1)

$\frac{c}{a}=1=\frac{1}{1}$ ---(2)

By comparing (1) and (2)

Clearly $a=1,c=1$ and $b=-4$

$\Rightarrow$ Quadratic polynomial is,

$a{x}^2+bx+c=x^2-4x+1$

Alternate method:

Sum of zeros $=\alpha +\beta =4$

Product of zeros $=\alpha \beta =1$

The Quadratic polynomial whose zeros are $\alpha$ and $\beta$ is

$x^2-(\alpha +\beta )x+\alpha \beta$

$=x^2-4+1$

Hence, the required Quadratic polynomial is $x^2-4+1$.