Alpha, Beta And Gamma Are The Zeroes Of Cubic PolynomialP(x)=ax3+bx2+cx+d,(Are Not Equal To 0) Then Product Of Their Zeroes [α.β.γ] Is

Sol:

Given:

\(\begin{array}{l}P(x) = ax^{3} + bx^{2} + cx + d\end{array} \)
…………………..[1]

Where

\(\begin{array}{l}a\neq 0\end{array} \)
is a cubic polynomial

If α,β,γ are the zeroes of the polynomial

then

\(\begin{array}{l}p(x) = (x – \alpha)(x – \beta) (x – \gamma)\end{array} \)

\(\begin{array}{l}p(x) = x^{3} – (\alpha +\beta +\gamma) x^{2} + (\alpha\beta + \beta\gamma + \gamma\alpha) x – (\alpha +\beta +\gamma)\end{array} \)
…………………..[2]

Now both the equations (1) and (2) are same. By comparing the coefficients, we get,

\(\begin{array}{l}\frac{a}{1} = \frac{b}{-\alpha -\beta -\gamma} = \frac{c}{\alpha \beta + \beta \gamma + \alpha}=\frac{d}{-\alpha\beta\gamma}\end{array} \)

On solving we get,

\(\begin{array}{l}\alpha +\beta +\gamma = \frac{-b}{a}\end{array} \)
= Sum of the roots

\(\begin{array}{l}\alpha\beta +\beta\gamma +\gamma\alpha = \frac{c}{a}\end{array} \)

\(\begin{array}{l}\alpha\beta\gamma = \frac{-d}{a}\end{array} \)
= Product of the roots

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