Question

The quadratic polynomial p(x) with -81 and 3 as product and one of zeroes respectively is

Answer 1

Suppose other zero of the polynomial is $\alpha$.
And one of the zero is 3.
Then product of the zeroes = $3\alpha =-81$
$\Rightarrow \alpha =\frac{-81}{3}=-27$
So other zero of the polynomial is -27.
Also sum of the zeroes of the polynomial = $-27+3=-24$
So the expression of required polynomial p(x) is given by;
$$\begin{gathered} x^2-\left(\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{zeroes}\mathrm{of}\mathrm{the}\mathrm{polynomial}\right)x+\left(\mathrm{product}\mathrm{of}\mathrm{the}\mathrm{zeroes}\right) \\ =x^2-\left(-24\right)x+\left(-81\right) \\ =x^2+24x-81 \end{gathered}$$
Therefore the quadratic polynomial $p\left(x\right)\mathrm{is}{x}^2+24x-81$