MT-2 Question - None

Question

If α and β are the zeros of the polynomial f(x) = x² + px + q, then a polynomial having $\frac{1}{\alpha} \text { and } \frac{1}{\beta}$ is its zero is

Accepted Answer

Let $\alpha$ and $\beta$ be the zeros of the polynomial $f(x)=x^{2}+p x+q$.Then,
$\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $=-\frac{p}{1}$$=-p$
And $\alpha \beta=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$ $=\frac{q}{1}$ = q
Let S and R denote respectively the sum and product of the zeros of a polynomial whose zeros are $\frac{1}{\alpha}$ and $\frac{1}{\beta}$, then
$S=\frac{1}{\alpha}+\frac{1}{\beta}$ $=\frac{\alpha+\beta}{\alpha \beta}$ $=\frac{-p}{q}$
$R=\frac{1}{\alpha} \times \frac{1}{\beta}$ $=\frac{1}{\alpha \beta}$ $=\frac{1}{q}$
Hence, the required polynomial $g(x)$ whose sum and product of zeros are S and R is given by
$x^{2}-S x+R=0$
$x^{2}+\frac{P}{q} x+\frac{1}{q}=0$
$\frac{q x^{2}+P x+1}{q}=0$
$\Rightarrow \quad q x^{2}+p x+1$
So $g(x)=q x^{2}+p x+1$

Question:

Step-by-Step Solution

Master MT-2 with Mathbee