Question

if p and q are the zeros of polynomial 3x² -5x + 2,write the polynomial in 'x' whose zeros are 1/p and 1/q​

Answer 1

Answer:

$\large \bold\red{2 {x}^{2} - 5x + 3}$

Step-by-step explanation:

Given that,

A Polynomial

$3 {x}^{2} - 5x + 2 = 0$

Also,

The roots of the Polynomial are p and q.

Now,

Comparing the coefficient of the given Polynomial with general form of a quadratic equation,

$\bold{a {x}^{2} + bx + c = 0}$

We have,

$\bold{a = 3 ,\;\: b = - 5 ,\;\:and\;\:c = 2}$

Also,

We know that,

$Sum \: of \: roots = - \frac{b}{a} = - ( \frac{ - 5}{3} ) = \frac{5}{3} \\ And \\ Product \: of \: roots = \frac{c}{a} = \frac{2}{3}$

Therefore,

We have,

$\bold{p + q = \frac{5}{3} }\: \: \: \: \: \: \: ..................(i) \\ and \\\bold{ pq = \frac{2}{ 3} } \: \: \: \: \: \: \: \: \: \: \: \: \: .................(ii)$

Now,

We have to find the quadratic Polynomial having roots $\bold{\frac{1}{p}\; and\;\frac{1}{q}}$

Therefore,

We have,

$Sum \: of \: roots = \frac{1}{p} + \frac{1}{q} = \frac{p + q}{pq}$

Putting the valuea from eqn (i) and (ii),

We get,

$= > Sum \: of \: roots = \frac{ \frac{5}{3} }{ \frac{2}{3} } = \frac{5}{2}$

And,

$Product \: of \: roots = \frac{1}{p} \times \frac{1}{q} = \frac{1}{pq}$

Putting the values from eqn (ii),

We get,

$= > Product \: of \: roots = \frac{1}{ \frac{2}{3} } = \frac{3}{2}$

Now,

We know that,

The Equation of a quadratic Polynomial is given as,

${x}^{2} - (sum \: of \: roots)x + (product \: of \: roots) = 0$

Therefore,

Putting the respective values,

We get,

$= > {x}^{2} - \frac{5}{2} x + \frac{3}{2} = 0 \\ \\ = > 2 {x}^{2} - 5x + 3 = 0$

Hence,

The required quadratic Polynomial is,

$\large \bold{2 {x}^{2} - 5x + 3}$