Question

if alpha beta and gamma are the zeroes of the polynomial 2x³+x²+13x+6 then evaluate alpha×beta+beta×gamma+gamma×alpha​

Answer 1

The value of αβ + βγ + γα is :

$\bf \dfrac{13}{2}\qquad=\qquad 6.5$

Step-by-step explanation :

Given -

α, β and γ are the zeroes of the polynomial 2x³ + x² - 13x + 6

To Find -

The value of αβ + βγ + γα.

Solution :

We know that :

$\bigstar\;\boxed{\bf \alpha\beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}}$

So,

$\implies \sf p(x)=2x3+x2-13x+6\\ \\ \\\implies \sf \dfrac{13}{2}\qquad=\qquad \bf 6.5$

Hence,

The value of αβ + βγ + γα is :

$\bf \dfrac{13}{2}\qquad=\qquad 6.5$

$\rule{300}{1.5}$

$\bigstar\;\textbf{\underline{\underline{Extra\;Formuale :-}}}\\ \\ \\\boxed{\begin{minipage}{7cm}{\underline{\underline{{\textbf {For cubic polynomial :}}}}}\\ \\ \sf 1)\; \alpha + \beta + \gamma = \dfrac{-b}{a} \\ \\ 2)\;\alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a} \\ \\ 3)\;\alpha \beta \gamma = \dfrac{-d}{a} \end{minipage}}$

$\boxed{\begin{minipage}{7cm}{\underline{\underline{{\textbf {For quadratic polynomial :}}}}}\\ \\ \sf 1)\;\alpha + \beta = \dfrac{-b}{a} \\ \\ 2) \;\alpha \beta = \dfrac{c}{a} \end{minipage}}$