Question

Find a quadratic polynomial whose zeroes are $2$ and $-5$ respectively.

Answer 1

Step $1$: Compute the sum of zeroes and product of zeroes.

Let us assume the quadratic polynomial be $a{x}^2+bx+c=0$, where $a\ne 0$ and its zeroes be $\alpha$ and $\beta$.

Given: Quadratic polynomial have zeroes $2$ and $-5$.

$\alpha =2,\beta =-5$

Sum of zeroes $=\alpha +\beta$

$$\begin{gathered} =2-5 \\ =-3 \end{gathered}$$

Product of zeroes $=\alpha .\beta$

$$\begin{gathered} =2(-5) \\ =-10 \end{gathered}$$

Step $2$: Compute the required equation.

We know that the quadratic equation in terms of sum and product of zeroes is given by, $k\left[x^2-\left(\alpha +\beta \right)x+\alpha \beta \right]$

where $k$ is a constant.

From the above calculations,

$\Rightarrow k\left[x^2+3x-10\right]$

When $k=1$ the quadratic equation will become,

$\Rightarrow x^2+3x-10$

Hence the quadratic polynomial whose zeroes are $2$ and $-5$ respectively is $x^2+3x-10$.