Question

How do you determine the sum and product of the roots of $3{{x}^{2}}+2x=0$?

Answer 1

To find the sum and product of the roots of the given polynomial equation $3{{x}^{2}}+2x=0$ , we firstly have to convert the given polynomial equation in the form of $a{{x}^{2}}+bx+c=0$ , where a = 1 and then the negative value of coefficient of $x$ is the sum of the roots of the polynomial equation and the constant term in the resultant polynomial equation is the product of the roots of polynomial equation .

Complete step-by-step solution:
let us now see to the above given question
$\Rightarrow$ $3{{x}^{2}}+2x=0$
To take this equation to the general form like, $a{{x}^{2}}+bx+c=0$ , we have to divide whole equation by $3$on LHS and RHS
Then after dividing we get,
$\Rightarrow$ $\dfrac{3}{3}{{x}^{2}}+\dfrac{2}{3}x+0=0$
$\Rightarrow$ ${{x}^{2}}+\dfrac{2}{3}x+0=0$
From the above explanation we can say that the sum of the roots is the negative value of the coefficient of the resultant normalized equation, that is sum of the roots of given polynomial equation is $-\dfrac{2}{3}$, and the product of the roots are constant value of the resultant normalized polynomial equation, that is $0$.

Note: Evaluating and finding the sum and product of the roots of a particular polynomial equation is done by the above solved method, one can find difficulty in converting the given polynomial equation to generalized normal form of polynomial equation, one should take care while dividing or multiplying the numbers to generalize the equation, and one may get confused if the constant or coefficient is zero, then the product or sum will also be zero.