Question

find a quadratic polynomial each with the given number as the sum and product of its zeroes respectively :
$- \frac{1}{3} \: \frac{1}{3}$


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Answer 1

Answer:

${\large{\pmb{\sf{★Given...}}}}$

Sum of Zeroes = -1/3

Product of Zeroes = 1/3

${\large{\pmb{\sf{★To \: Find...}}}}$

Quadratic Polynomial...?

$\: {\large{\pmb{\sf{★Used \: Formula...}}}}$

$\boxed{ \sf{General \: Form =k \bigg[ {x}^{2} - ( \alpha + \beta )x + \alpha \beta \bigg] }}$

${\large{\pmb{\sf{★Solution...}}}}$

From given, α + β = -1/3 , α β = 1/3

Substitute in formula,

$: { \implies { \sf{k \bigg[ {x }^{2} - \bigg( \frac{ - 1}{3} \bigg)x + \frac{1}{3} \bigg]}}}$

$\: : { \implies { \sf{k \bigg[ {x }^{2} + \frac{1}{3} x + \frac{1}{3} \bigg]}}}$

By Taking LCM,

$\: : { \implies { \sf{k \bigg[ \frac{3 {x}^{2} + x + 1 }{3} \bigg]}}}$

Now Taking k = 3,

$\: : { \implies { \sf{3 \bigg[ \frac{3 {x}^{2} + x + 1 }{3} \bigg]}}}$

$\: \: : { \implies { \sf{ \cancel{3} \bigg[ \frac{3 {x}^{2} + x + 1 }{ \cancel{3}} \bigg]}}}$

$: { \implies{ \bf{3 {x}^{2} + x + 1}}}$

${\large{\pmb{\sf{★Final Answer...}}}}$

Therefore, 3x² + x + 1 is our required polynomial.