Question

If alpha and beta are zeros of xsquare + 4x +3 write the polynomial whose zero are (1+B/alpha) and (1+alpha/beta)???​

Answer 1

Correct Question -

If alpha and beta are zeros of the following polynomial - ${x}^2 + 4x +3$ write the polynomial whose zeroes are $\dfrac{ 1 + \beta }{ \alpha }$ and $\dfrac{1 + \alpha }{ \beta }$

Solution -

We have to write the polynomial whose zeroes are $\dfrac{ 1 + \beta }{ \alpha }$ and $\dfrac{1 + \alpha }{ \beta }$

So,

$\sf{ Sum \: Of \: Zeroes \: :- } = \dfrac{ \alpha + \beta + 2 \alpha \beta }{ \alpha \beta }$

$\sf{ In \: the \: given \: polynomial \:: } {x}^2 + 4x +3 \\ \\ \sf{ Sum Of Zeroes \::- \dfrac{ -b}{a} = -4 } \\ \\ \sf{ Product Of Zeroes \::- \dfrac{ c}{a} = 3 } \\ \\ So, \\ \\ \: \alpha + \beta = -4 \\ \\ \alpha \beta = 3 \\ \sf{ So \: Sum \: Of \: Zeroes \: :- } = \dfrac{ \alpha + \beta + 2 \alpha \beta }{ \alpha \beta } \\ \\ = \dfrac{ -4 + 6 }{ 3 } \\ \\ => \dfrac{2}{3}$

$\sf{ We \: know \: that \: - } \\ \\ \sf{ A \: polynomial \: can \: be \: written \: as \: - } \\ \\ {x}^2 - sx + p = 0 \\ \\ \sf{ Where \: - } \\ \\ \sf{ s \: is \: The \: Sum \: Of \: Roots } \\ \\ \sf{ p \: is \: The \: Product \: Of \: Roots } \\ \\ \tt{ Substituting \: the \: required \: values \: - } \\ \\ => { x } ^ 2 - \dfrac{2x}{3} + 3 = 0 \\ \\ => 3 { x } ^ 2 - 2x + 9 = 0 .............. [ A ]$