Question

Write a polynomial $p\left(x\right)$, whose sum and product of the zeroes are $\frac{9}{4}$and $\frac{5}{8}$respectively

Answer 1

Finding the required polynomial:

Sum of zeroes $=\alpha +\beta =\frac{9}{4}$
Product of zeroes $=\alpha \beta =\frac{5}{8}$

Let the required polynomial be $p\left(x\right)$.

The standard quadratic polynomial, $p\left(x\right)=x^2-(\alpha +\beta )x+\alpha \beta$,

Therefore,

$$\begin{gathered} p\left(x\right)=x^2-\frac{9}{4}x+\frac{5}{8} \\ \therefore p\left(x\right)=8x^2-18x+5 \end{gathered}$$

Hence, the required polynomial is $p\left(x\right)=8x^2-18x+5.$