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

Finding the quadratic polynomial .

Let $\alpha$ and $\beta$ be the roots of a quadratic equation.

Sum of the zeroes

$\Rightarrow \alpha +\beta =\frac{1}{4}$

Product of the zeroes

$\Rightarrow \alpha \beta =-1$

Substitute these values in the quadratic equation $\mathit{k}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\left(\alpha +\beta \right)\mathit{x}\mathbf{+}\mathit{\alpha }\mathit{\beta }\mathbf{)}$, where $k$ is any non-zero number.

$$\begin{gathered} =k\left(x^2-\frac{1}{4}x-1\right) \\ =\frac{k}{4}\left(4x^2-x-4\right) \\ =m\left(4x^2-x-4\right)\mathbf{[}\mathbf{\text{we can assume}}\mathbf{}\mathbf{k}\mathbf{/}\mathbf{4}\mathbf{}\mathbf{\text{as}}\mathbf{}\mathbf{m}\mathbf{\ne }\mathbf{0}\mathbf{]} \end{gathered}$$

Hence, the quadratic equation is $\mathit{m}\left(4x^2-x-4\right)$.