preboard leelavati school
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Answer:
▪Key Concept :-
☆ Relationship b/w Zeros And Coefficients of a Cubic Polynomial
If α , β and γ are the Zeros of a Cubic Polynomial
of the Form:
\bf a{x}^{3} + b {x}^{2} + cx + dax
3
+bx
2
+cx+d
Then,
\begin{gathered} \bf \alpha + \beta + \gamma = - \frac { b}{a} \\ \\ \bf\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a} \\ \\ \bf \alpha \beta \gamma = - \frac{d}{a} \end{gathered}
α+β+γ=−
a
b
αβ+βγ+γα=
a
c
αβγ=−
a
d
___________________________
▪Solution :-
Here we have,
\bf\alpha + \beta + \gamma = - \frac{( - 2)}{2} = 1α+β+γ=−
2
(−2)
=1
\begin{gathered} \bf \alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{3}{2} \\ \\ \bf \alpha \beta \gamma = - \frac{ ( - 4)}{2} = 2\end{gathered}
αβ+βγ+γα=
2
3
αβγ=−
2
(−4)
=2
Now ,
\begin{gathered} \frac{ \alpha }{ \beta \gamma } + \frac{ \beta }{ \gamma \alpha } + \frac{ \gamma }{ \alpha \beta } \\ \\ = \frac{ { \alpha }^{2} + { \beta }^{2} + { \gamma }^{2} }{ \alpha \beta \gamma } \\ \\ \bf \bigg \{ \frac{ \large Add \: and \: Subtract}{ \underline{2( \alpha \beta + \beta \gamma + \gamma \alpha ) \: in \: Numerator } }\bigg \}\end{gathered}
βγ
α
+
γα
β
+
αβ
γ
=
αβγ
α
2
+β
2
+γ
2
{
2(αβ+βγ+γα)inNumerator
AddandSubtract
}
\begin{gathered} = \frac{( { \alpha + \beta + \gamma )}^{2} - 2( \alpha \beta + \beta \gamma + \gamma \alpha) }{ \alpha \beta \gamma } \\ \\ \bf \Large\{ Putting \: \: values \}\end{gathered}
=
αβγ
(α+β+γ)
2
−2(αβ+βγ+γα)
{Puttingvalues}
\begin{gathered} = \dfrac{ {(1)}^{2} - 2( \dfrac{3}{2} )}{2} \\ \\ = \frac{1 - 3}{2} \\ \\ = \frac{ - 2}{ \: \: \: 2} \\ \\ = \huge\purple{{ {\bf \: - 1}}}\end{gathered}
=
2
(1)
2
−2(
2
3
)
=
2
1−3
=
2
−2
=−1
\begin{gathered} \Large \red{\mathfrak{ \text{W}hich \: \: is \: \: the \: \: required} }\\ \huge \red{\mathfrak{ \text{ A}nswer.}}\end{gathered}
Which is the required
Answer.