Question

If alpha and beta are the zeroes of polynomial p(x) = x² - 7x + 10, find the quadratic polynomial with zeroes (-alpha) and (-beta).​

Answer 1

Answer:

I hope this solution is help you.

Answer 2

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♠ Given :-

$\large \pmb{\sf{\alpha \: and \: \beta \: are \: zeros \: of \:polynomial}} : \\ \bf{x}^{2} -7x + 10$

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♠ To Find :-

$\sf{A \: Quadratic \:polynomial \: whose \: zeros \: are}: \\ \bf{-\alpha \: and \: - \beta }$

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♠ Concept To Mind :-

$\pmb{\mathfrak{For \: \text{A} \: Qudratic \:pol \text{y}nomial \: of \: \: the \: \: Form}} :$$\Large\bf a {x}^{2} + bx +c \\ \\ \large\sf{Sum \: of \: Zeros = - \dfrac{b}{a} } \\ \\ \large \sf{ Product \: of \: Zeros = \frac{c}{a} }$

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♠ Solution :-

$\pmb{As \: {\alpha \: and \: \beta \: are \: zeros \: of \:polynomial}} : \\ \bf{x}^{2} -7x + 10$

$\large \bigstar \: \underline{ \pmb{ \mathfrak{ \text{H}ence , }}}$

$\large \pmb{ \alpha + \beta = 7} \: \: \: \: - - - (1)$

$\large \pmb{ \alpha \beta = 10} \: \: \: \: - - - (2)$

$\large \bigstar \: \underline{ \pmb{ \mathfrak{ Now, }}}$

For The Quadratic Polynomial having zeros - α and - β.

$\large\sf{Sum \: of \: Zeros =S= - \alpha - \beta } \\ \\ =\large - (\alpha+\beta)$

$\large \bigstar \: \underline{ \pmb{ \mathfrak{ Using \: \: (1 ) }}}$

$\large:\longmapsto\pmb{ \boxed{ \boxed{S = - 7}}}$

$\large\sf{Product \: of \: Zeros =P= (- \alpha ) (- \beta )} \\ \\ = \large\alpha\beta$$\large \bigstar \: \underline{ \pmb{ \mathfrak{ Using \: \: (2) }}}$

$\large:\longmapsto\pmb{ \boxed{ \boxed{P = 7}}}$

We Know,

Quadratic Polynomial having sum of Zeros S and Product of Zeros P have General Form :

x² - Sx + P

Hence Required Polynomial is : $\Large\pink{\pmb{{x}^{2} -7x + 10}}$

$\Large \red{\mathfrak{ \text{W}hich \: \: is \: \: the \: \: required} }\\ \huge \red{\mathfrak{ \text{ A}nswer.}}$

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