Question

Quadratic polynomial 2x^2 - 3x + 1 has zeroes alpha and beeta. Now form a quadratic polynomial whose zeroes are 1/alfa and 1/beeta

Answer 1

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☯️Quadratic polynomial 2x^2 - 3x + 1 has zeroes alpha and beeta. Now form a quadratic polynomial whose zeroes are 1/alfa and 1/beta..

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Polynomial= x2-3x+2

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$: \large{ \sf{ \implies{let \: p(x) = 2 {x}^{2} - 3x + 1}}} \\ \\ \large{ \sf{ \star{ \: we \: know \: if \: \alpha \: and \: \beta \: are \: zeroes \: then}}} \\ \\ \large{ \sf{ \red{ \alpha + \beta = \frac{ - coefficient \: of \: x}{coefficient \: of \: {x}^{2} } }}} \\ \\ \large{ \sf{ \red{ \alpha \beta = \frac{constant \: term}{coefficient \: of \: {x}^{2} } }}} \\ \\ \\ : \large{ \sf{ \implies{ \alpha + \beta = \frac{ - ( - 3)}{2} }}} = \frac{3}{2} \\ \\ : \large{ \sf{ \implies{ \alpha \beta = \frac{1}{2} }}} \\ \\ \\ \large{ \sf{we \: have \: to \: find \: polynomial \: with \: zeroes \: \frac{1}{ \alpha } and \: \frac{1}{ \beta } }} \\ \\ \large{ \sf{ \mapsto{ \purple{u \: must \: know..for \: a \: polynomial}}}} \\ \\ \large{ \sf{ \boxed{ \bold{ \green{ {x}^{2} - (sum \: of \: zeroes)x + product \: of \: zeroes}}}}} \\ \\ \\ \large{ \sf{ \mapsto{ \purple{by \: using \: this \: formula..}}}} \\ \\ : \large{ \sf{ \implies{polynomial = {x}^{2} - ( \frac{1}{ \alpha } + \frac{1}{ \beta } ) + \frac{1}{ \alpha } \times \frac{1}{ \beta } }}} \\ \\ \large{ \sf{ = {x}^{2} - ( \frac{ \alpha + \beta }{ \alpha \beta })x + \frac{1}{ \alpha \beta } }} \\ \\ \large{ \sf{ \green{putting \: values \: of \: \alpha + \beta \: and \: \alpha \beta }}} \\ \\ \large{ \sf{ = {x}^{2} - ( \frac{ \frac{3}{ \cancel{2}} }{ \frac{1}{ \cancel{2}} } )x + \frac{1}{ \frac{1}{2} } }} \\ \\ \large{ \sf{ \boxed{ \red{ = {x}^{2} - 3x + 2}}}}$

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$\large{ \bold{ \underline{ \pink{required \: polynomial \: is \: {x}^{2} - 3x + 2}}}}$

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✒The zero of the polynomial is defined as any real value of x, for which the value of the polynomial becomes zero. A real number k is a zero of a polynomial p(x), if p(k) = 0.