Question

If we are given that $\alpha ,\beta ,\gamma $ are the zeroes of cubic polynomial $3{{x}^{3}}-2{{x}^{2}}+5x-6=0$ then find
( a ) $\alpha +\beta +\gamma $
( b ) $\alpha \cdot \beta +\beta \cdot \gamma +\gamma \cdot \alpha $

Answer 1

Hint: We will use the results based on of relationship between roots of cubic equation to evaluate the values of $\alpha +\beta +\gamma $ and $\alpha \cdot \beta +\beta \cdot \gamma +\gamma \cdot \alpha $ which are given by the relation of coefficients of cubic equation such as $\alpha +\beta +\gamma =-\dfrac{b}{a}$ and $\alpha \cdot \beta +\beta \cdot \gamma +\gamma \cdot \alpha =\dfrac{c}{a}$ for cubic equation $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$ where $a\ne 0.$

Complete step-by-step solution:
Now, firstly we will find the coefficients of ${{x}^{3}}$ ,${{x}^{2}}$ ,$x$ and constant d from the given polynomial $3{{x}^{3}}-2{{x}^{2}}+5x-6=0$ by comparing it with the general form of cubic polynomial which is expressed as $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$.
On comparing given polynomial with general form of cubic polynomial, we get coefficients of cubic polynomial $3{{x}^{3}}-2{{x}^{2}}+5x-6=0$equals to,
 a = 3, b = -2, c = 5, d = -6
Now,
( a ) Here, we know that $\alpha +\beta +\gamma =-\dfrac{b}{a}$……( i ),
So, we can obtain the value of $\alpha +\beta +\gamma $ easily by substituting the values of b and a in an equation ( i )
Substituting values of a = 3 and b = -2 in$\alpha +\beta +\gamma =-\dfrac{b}{a}$, we get
$\alpha +\beta +\gamma =-\dfrac{(-2)}{3}$
On simplifying signs, we get
$\alpha +\beta +\gamma =\dfrac{2}{3}$
( b ) Here, we know that $\alpha \cdot \beta +\beta \cdot \gamma +\gamma \cdot \alpha =\dfrac{c}{a}$…..( ii ),
So, we can obtain the value of $\alpha \cdot \beta +\beta \cdot \gamma +\gamma \cdot \alpha $easily by substituting the values of c and a in an equation ( ii )
Substituting values of a = 3 and c= 5 in $\alpha \cdot \beta +\beta \cdot \gamma +\gamma \cdot \alpha =\dfrac{c}{a}$, we get
$\alpha \cdot \beta +\beta \cdot \gamma +\gamma \cdot \alpha =\dfrac{5}{3}$.
Hence, the values of $\alpha +\beta +\gamma $ and $\alpha \cdot \beta +\beta \cdot \gamma +\gamma \cdot \alpha $ are equals to $\dfrac{2}{3}$ and $\dfrac{5}{3}$ respectively .

Note: Remember these formulae as they are very helpful in solving questions. While calculating the coefficients of a cubic equation, try to avoid signs error as this makes the answer incorrect. Simplification of signs should be done carefully.