Question

If alpha and beta are thr zeroes of the polynomial px=x^-px+q then find the value of 1÷alpha+1÷beta

Answer 1

Answer: $\bold{\frac{p}{q}}$


The polynomial given is:

$p(x) = x^2-px+q$

The zeros of the polynomial are $\alpha$ and $\beta$

So, 

Sum of zeros is:

$\alpha + \beta = -\frac{\text{Coefficient of x}}{\text{Coefficient of }x^2} \\ \\ \\ \implies \alpha + \beta = -\frac{-p}{1} \\ \\ \\ \implies \alpha + \beta = p$


Also,

Product of Zeros is:

$\alpha \beta = \frac{\text{Constant Term}}{\text{Coefficient of }x^2} \\ \\ \\ \implies \alpha \beta = \frac{q}{1} \\ \\ \\ \implies \alpha \beta = q$


Now, we can find the required value:


$\frac{1}{\alpha} + \frac{1}{\beta} \\ \\ \\ = \frac{\alpha+\beta}{\alpha\beta} \\ \\ \\ = \frac{p}{q} \\ \\ \\ \implies \boxed{\frac{1}{\alpha}+\frac{1}{\beta}=\frac{p}{q}}$


Thus, The answer is  $\bold{\frac{p}{q}}$