Question

find the condition that zeros of polynomial p(x)=ax2+bx+c are reciprocal of each other

Answer 1

Hence the condition when zeroes of the polynomial are the reciprocal to each other is a= c

Hope this will help you.....

Answer 2

Answer:

It is true that zeros of polynomial are reciprocal to each other at the condition a = c.

To find:

Zeros of polynomial $p(x)=ax^2+bx+c$ are reciprocal of each other.

Solution:

One zero of the polynomial $ax^2+bx+c$ is reciprocal of the other.

Assume that one of the zero of above polynomial as x, then another zero will be 1/x.

Product of zeroes  

$\begin{aligned} = ( x ) \left( \frac { 1 } { x } \right) & = \frac { \text { constant } } { x ^ { 2 } \text { coefficient } } \\\\ 1 & = \frac { c } { a } \\\\ a & = c \end{aligned}$

Let us take one polynomial to find that when a = c, zeros are reciprocal.

$\begin{array} { l } { 4 \mathrm { x } ^ { 2 } + 10 \mathrm { x } + 4 = 0 } \\\\ { 4 \mathrm { x } ^ { 2 } + 8 \mathrm { x } + 2 \mathrm { x } + 4 = 0 } \\\\ { 4 \mathrm { x } ( \mathrm { x } + 2 ) + 2 ( \mathrm { x } + 2 ) = 0 } \\\\ { ( \mathrm { x } + 2 ) ( 4 \mathrm { x } + 2 ) = 0 } \end{array}$

First zero = x + 2 i.e. x = -2

Second zero = 4x + 2 i.e. 4x = -2 then x = -1/2

Hence, it can be said that a = c, then zeros are reciprocal.