Question

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1.If α and β are zeroes of the polynomial x²-2x-15 then form a quadratic polynomial whose zeroes 2α and 2β.

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Answer 1

Given : If α and β are zeroes of the polynomial x²-2x-15

To Find : A quadratic polynomial whose zeroes 2α and 2β ?

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❍ First Find out all Cofficients of Given Polynomial x²-2x-15.

• Cofficients of x² = 1

• Cofficients of x = -2

• Cofficients Term = -15

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$\leadsto$ α & β are two zeroes of Polynomial x²-2x-15.

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$\underline{\frak{As~ we ~know~ that~:}}$

• $\boxed{\sf\pink{ \alpha + \beta = \dfrac{ - Cofficients \: of \: x}{cofficients \: of \: {x}^{2} }}}$★

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$\underline{\bf{Now~ By ~Substituting ~the ~Cofficients~:}}$

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$~~~~~~~~~~{\sf:\implies{\alpha + \beta = \dfrac{ - ( - 2)}{1} }}$

$~~~~~~~~~~{\sf:\implies{\alpha + \beta = \dfrac{2}{1}}}$

Or,

$~~~~~~~~~~{\sf:\implies{\boxed{\frak{\pink{\alpha + \beta~ = ~2}}}}}$★

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$\underline{\frak{As ~we ~know~ that~:}}$

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$~~~~~~~~~~{\sf:\implies\pink{\alpha × \beta ~=~\dfrac{Constant ~Term}{Cofficient~of~x^2}}}$

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$\underline{\bf{Now ~By ~Substituting~ the ~Cofficients~:}}$

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$~~~~~~~~~~{\sf:\implies{\alpha × \beta~=~\dfrac{-15}{1}}}$

Or,

$~~~~~~~~~~{\sf:\implies{\underline{\boxed{\frak{\pink{\alpha × \beta~=~-15}}}}}}$★

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$\underline{\frak{As~ we ~know~ that~:}}$

• Quadratic Polynomial = x² - (Sum of zeroes) x + Product of zeroes.

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Given that,

• The zeroes of new formed Polynomial will be 2α & 2β.

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$\underline{\bf{Now~ By ~Substituting ~the ~Given ~Values~:}}$

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$~~~~~~~~~~{\sf:\implies{\bigg(x^2~-~(2\alpha~ + ~2\beta)x~+~(2\alpha~+~2\beta)\bigg)}}$

$~~~~~~~~~~{\sf:\implies{\bigg(x^{2}~-~2(\alpha~+~\beta)x~+~(2\alpha~+~2\beta)\bigg)}}$

$~~~~~~~~~~{\sf:\implies{\bigg(x^{2}~-~2(\alpha~+~\beta)x~+~4\alpha\beta\bigg)}}$

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$\underline{\frak{As ~we~ have ~found~:}}$

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$~~~~~~~~~~{\sf:\implies{\alpha~×~\beta~=~-15}}$

$~~~~~~~~~~{\sf:\implies{\alpha~+~\beta~=~2}}$

$~~~~~~~~~~{\sf:\implies{\bigg(x^{2}~-~2(2)x~+~4(-15)\bigg)}}$

$~~~~~~~~~~{\sf:\implies{\bigg(x^{2}~-~4x~-~60\bigg)}}$

Or,

• ${\underline{\boxed{\frak{\pink{New~ Formed ~Quadratic~ Polynomial ~=~x^{2}~-~4x~-~60}}}}}$

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Hence,

$\therefore\underline{\sf{New~ Formed ~Quadratic ~Polynomial~=~\bf{x^{2}~-~4x~-~60}}}$