Question

From the polynomial whose
zeroes are
3+√3 and 3-√3​

Answer 1

$\large \sf {\underline {\underline{ \color{purple}{Question : - }}}}$

$\sf{Form \: the \: polynomial \: whose \: zeroes \: are \: 3 + \sqrt{3} \: and \: 3 - \sqrt{3} }$

$\large\sf {\underline {\underline{ \color{purple}{ Answer: - }}}}$

$\sf{ : \implies{x}^{2}  - 6x + 6}$

$\large\sf {\underline{ \color{red}{Given \: that : - }}}$

$\sf{The \: zeroes \: are \: 3 + \sqrt{3} \: and \: 3 - \sqrt{3} . }$

$\sf{∴ \: Sum \: of \: the \: zeroes \: = \: ( 3 + \sqrt{3} ) + ( 3 - \sqrt{3} ) = 6 }$

$\circ \: \: \sf {\underline{ \color{green}{Their \: Product : }}}$

$\sf{ : \implies \: Product \: of \: the \: zeroes \: = \:  ( 3 +  \sqrt{3} )  ( 3 -  \sqrt{3} )}$

$\sf{ : \implies \: = 9 - {( \sqrt{3} )}^{2} }$

$\sf{ : \implies \: \: = \: 9 - 3}$

$\sf{ : \implies \:= { \boxed{ \color{red}6}}}$

$\sf{The \: required \: polynomial \: is  : }$

$\sf{ {x}^{2} \: - \: (sum \: of \: the \: zeroes) \: x \: + \: Product \: of \: the \: zeroes \: ⇒ {x}^{2} - (6)x + 6 }$

${ \underline{ \boxed{ \sf{ \pink{ : \implies \: ∴ \: {x}^{2} - 6x + 6 }}}}}$

Answer 2

The zeroes are 3 + √3 and 3 - √3. ∴ Sum of the zeroes = ( 3 + √3) + ( 3 - √3) = 6

∴ Product of the zeroes = ( 3 + √3) ( 3 - √3) = 9 - (√3)2

= 9 - 3 = 6

The required polynomial is

x2 - (sum of the zeroes) x + Product of the zeroes ⇒ x2 - (6)x + 6

∴ x2 - 6x + 6