Question

Find a quadratic polynomial whose zeroes are √3 + 1 and √3 − 1

Answer 1

Answer:

if two zeros of a quadratic polynomial is given, then polynomial can be formed as,

x² - x( sum of zeros ) + product of zeros

=> x² - x ( √3+1+√3-1) + [(√3+1)(√3-1)]

=> x² - x(2√3) + (3-1)

=> x² - 2√3x + 2

Answer 2

The quadratic polynomial whose zeroes are √3 + 1 and √3 − 1 is x² - 2√3x + 2

Given :

The zeroes √3 + 1 and √3 − 1

To find :

The quadratic polynomial

Concept :

If the Sum of zeroes and Product of the zeroes of a quadratic polynomial is given then the quadratic polynomial is

$\sf{ {x}^{2} -(Sum \: of \: the \: zeroes )x + Product \: of \: the \: zeroes }$

Solution :

Step 1 of 3 :

Write down the given zeroes

Here it is given that zeroes are √3 + 1 and √3 − 1

Step 2 of 3 :

Find Sum of zeroes and Product of the zeroes

Sum of zeroes

$\sf = ( \sqrt{3} + 1 ) + ( \sqrt{3} - 1 )$

$\sf = \sqrt{3} + 1 + \sqrt{3} - 1$

$\sf = 2 \sqrt{3}$

Product of the zeroes

$\sf = ( \sqrt{3} + 1 ) ( \sqrt{3} - 1 )$

$\sf = {( \sqrt{3} )}^{2} - {1}^{2}$

$\sf = 3 - 1$

$\sf = 2$

Step 3 of 3 :

Find the quadratic polynomial

Hence the required quadratic polynomial

$\displaystyle \sf = {x}^{2} -(Sum \: of \: the \: zeroes )x + Product \: of \: the \: zeroes$

$\displaystyle \sf{ = {x}^{2} - (2 \sqrt{3} )x + 2}$

$\displaystyle \sf{ = {x}^{2} - 2 \sqrt{3} x + 2}$