Question

if alfha and beta are zeroes of polynomial x^2-2x-8 form a quadratic polynomial whose zeroes are 3alfha and 3beta
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Answer 1

$\huge{\mathfrak{\overline{\underline{\underline{\blue{Answer}}}}}}$

$\huge\bold{Given}$

Given α and β be the zeroes of polynomials,

Quadratic equation ${\green{\overline{\green{\underline{\blue{\boxed{\orange{\mathtt{x² - 2x - 8}}}}}}}}}$

Sum Of the zeroes

α + β = $\frac{-b}{a}$

⠀⠀ = $\frac{-(-2)}{1}$

⠀⠀ ⠀ = 2

Product if zeroes

αβ = $\frac{c}{a}$

⠀ = $\frac{-8}{1}$

⠀ = - 8

Now :

They given 3α and 3β

Sum of 3α and 3β is = 3α + 3β

⠀ ⠀ ⠀ ⠀ ⠀ ⠀ ⠀ = 3 ( α + β )

⠀ ⠀ ⠀ ⠀ ⠀ ⠀ ⠀ = 3 ( 2 )

⠀ ⠀ ⠀ ⠀ ⠀ ⠀ ⠀ = 6

Product of 3α and 3β is = 3αβ × 3αβ

⠀ ⠀ ⠀ ⠀ ⠀ ⠀ ⠀ ⠀ = 9 ( - 8 )

⠀ ⠀ ⠀ ⠀ ⠀ ⠀ ⠀ ⠀ = - 72

Now substitute the value in

x² + ( sum of zeroes ) x + product of zeroes

==> x² + 6x - 72 = 0

⠀ ⠀ ⠀ ⠀ ⠀ ⠀ ⠀

Answer 2

${\huge{\mathrm{\star\:\:Required\:Answer \:\:\star}}}$

$\ast\mathrm{Given : }$

${\alpha \:and \:\beta}$ are zeroes of polynomial.

$\mathtt{ polynomial : {x}^{2} - 2x - 8}$

$\ast\mathrm{To \:form : }$

Quadratic polynomial whose zeroes : ${3\alpha \:and\: 3\beta\:}$

$\ast\mathrm{ Solution : }$

${\longmapsto{\mathtt{ {x}^{2} - 2x - 8 }}}$

${\longmapsto{\mathtt{\alpha + \beta = 2}}}$

${\longmapsto{\mathtt{ \alpha\beta = -8 }}}$

${\longmapsto{\mathtt{ So , 3\alpha + 3\beta = 3(\alpha + \beta) }}}$

${\longmapsto{\mathtt{ 3(\alpha + \beta) = 3 ( 2 ) = 6 }}}$

${\longmapsto{\mathtt{ Then , 3\alpha × 3\beta = 9\alpha\beta }}}$

${\longmapsto{\mathtt{ 3(\alpha\beta ) = 9( -8 ) = -72 }}}$

${\ast{\mathrm{ Thus , \:required\: quadratic\: polynomial\:is\::}}}$

${\longmapsto{\mathtt{ {x}^{2} - 3(\alpha + \beta)x + 9(\alpha\beta) }}}$

${\longmapsto{\mathtt{ {x}^{2} - 3(2)x + 9(-8 ) }}}$

${\green{\overline{\green{\underline{\purple{\boxed{\mathtt{\orange{ {x}^{2} - 6x -72 }}}}}}}}}$

⠀⠀⠀⠀⠀⠀⠀ ⠀⠀_____________________

⠀⠀${\blue{\boxed{\mathbb{\green{Brainly\:\:Booster}}}}}$ ⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀ ⠀

Finding zeroes of the above quadratic polynomial :

$\mathrm{{x}^{2} \:-\: 6x \:- \:72 }$

⠀⠀By factorizing the middle term

$\mathrm{{x}^{2}\: - \:( 12 - 6 )x\: - \:72 }$

$\mathrm{{x}^{2}\:-12x \:+\:6x\:-\:72 }$

$\mathrm{x(x - 12 ) \: +\:6(x - 12) }$

$\mathrm{(x - 12 )(x + 6) }$

$\mathrm{For \:(x - 12)\::}$

$\mathrm{x\: - \:12 \:= \:0 }$

$\mathrm{ x \:= \:12}$

$\mathrm{For \:(x + 6)\::}$

$\mathrm{x\: + \:6 \:= \:0 }$

$\mathrm{ x \:= \:-6}$

${\blue{\boxed{\green{\mathrm {value \:of\:x\:is\: 12 \:or\:-6}}}}}$

⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀ ⠀

________________________

$\ast$Quadratic polynomial : The polynomial having highest degree as 2 is called quadratic polynomial.

$\ast$When zero is added after polynomial , then it became quadratic equation.

$\ast$The quadratic equation has graph type : parabola .

$\ast$The general formula of quadratic equation is $\mathtt{a{x}^{2} + bx + c = 0 }$