Question

We are given that \[\alpha ,\beta \,\& \gamma \] are the zeroes of cubic polynomial \[P(x) = a{x^3} + b{x^3} + cx + d,(a \ne 0)\] then product of their zeroes \[\left[ {\alpha .\beta .\gamma } \right] = .....\]

Answer 1

Hint: To solve this type of problem first we will write general equation of cubic polynomial which is \[a{x^3} + b{x^2} + cx + d = 0\] and then we will make coefficient of \[x^3\] as 1 using division operation. Then we will write a cubic polynomial equation using zeros in the form of factors and then simplify and compare both equations to get the desired result.

Complete step-by- step solution:

General cubic polynomial can be written as -

\[p(x) = a{x^3} + b{x^2} + cx + d(a \ne 0)\]

Now cubic equation can be written as-

\[ \Rightarrow a{x^3} + b{x^2} + cx + d = 0\]

\[ \Rightarrow {x^3} + \dfrac{b}{a}{x^2} + \dfrac{c}{a}x + \dfrac{d}{a} = 0....(3)\]

Now,

If \[\alpha ,\beta ,\gamma \] are zeros then

\[(x - \alpha )(x - \beta )(x - \gamma ) = 0\]

On multiplying first two, we get:

\[({x^2} - (\alpha  + \beta )x + \alpha \beta )(x - \gamma ) = 0\]

On multiplying the result with \[(x - \gamma )\], we get:

\[{x^3} - (\alpha  + \beta ){x^2} + \alpha \beta x - \gamma {x^2} + \gamma (\alpha  + \beta )x  - \alpha \beta \gamma  = 0\]

On simplifying, we get:

\[{x^3} - (\alpha  + \beta  + \gamma ){x^2} + (\alpha \beta  + \beta \gamma  + \gamma \alpha ) x  - \alpha \beta \gamma  = 0..........(4)\]

Now compare of equation (3) and (4), product of roots \[\alpha \beta \gamma  =  - \dfrac{d}{a}\]

Note: In quadratic equation like this, 

\[a{x^2} + bx + c = 0(a \ne 0)\]

On dividing the equation throughout by ‘a’, we have:

\[ \Rightarrow {x^2} + \dfrac{b}{a} \times x  + \dfrac{c}{a} = 0.......(1)\]

Now if \[\alpha \] and \[\beta \] zeroes of the quadratic equation, then:

\[(x - \alpha )(x - \beta ) = 0\]

\[{x^2} - \alpha x - \beta x + \alpha b = 0\]

\[{x^2} - (\alpha  + \beta )x + \alpha b = 0....(2)\]

From equation 1 & 2, we have:

Sum of zeroes \[ \Rightarrow \alpha  + \beta  =  - \dfrac{b}{a}\]

Product of zeroes \[ \Rightarrow \alpha \beta  = \dfrac{c}{a}\]