Question

Find a quadratic polynomial if sum and product of its zeroes are 2√3 and 2 respectively.

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

Sum and product of zeroes of quadratic polynomial are 2√3 and 2.

Let assume that

f(x) be the required quadratic polynomial

and

$\rm \: \alpha , \: \beta \: be \: the \: zeroes \: of \: quadratic \: polynomial \: f(x)$

So, it is given that,

$\rm \: \alpha + \beta = 2 \sqrt{3}$

and

$\rm \: \alpha \beta = 2$

Now, the required Quadratic polynomial is given by

$\rm \: f(x) = k\bigg( {x}^{2} - ( \alpha + \beta )x + \alpha \beta\bigg) \: where \: k \: \ne \: 0$

So, on substituting the values, we get

$\rm \: f(x) = k\bigg( {x}^{2} - 2 \sqrt{3} x + 2\bigg) \: where \: k \: \ne \: 0$

Hence,

The required Quadratic polynomial is

$\boxed{\sf{ \:\rm \: f(x) = k\bigg( {x}^{2} - 2 \sqrt{3} x + 2\bigg) \: where \: k \: \ne \: 0 \: \: }}$

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Additional Information :-

For Cubic Polynomial

${ \rm\: \alpha , \beta , \gamma \: are \: zeroes \: of \: a {x}^{3} + b {x}^{2} + cx + d, \: then}$

$\boxed{ \bf{ \: \alpha + \beta + \gamma = - \dfrac{b}{a}}}$

$\boxed{ \bf{ \: \alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}}}$

$\boxed{ \bf{ \: \alpha \beta \gamma = - \dfrac{d}{a}}}$

For Quadratic polynomial

${\rm \: \alpha , \beta \: are \: zeroes \: of \: a {x}^{2} + b x + c, \: then}$

$\boxed{ \bf{ \: \alpha + \beta = - \dfrac{b}{a}}}$

$\boxed{ \bf{ \: \alpha \: \beta = \dfrac{c}{a}}}$