Question

What is the relationship between zeroes and co - efficients of a cubic polynomial ? Explain Mathematically .

Answer 1

General form of cubic polynomial of ax3 + bx 2+ cx + d where a ≠ 0. There are three zeroes of cubic polynomial.

The sum of zeroes of the cubic polynomial = $\frac{-b}{a}$ = $\frac{- coefficient of x^{2}}{coefficient of x^{3} }$

Sum of the product of zeroes taken two at a time = $\frac{c}{a}$ = $\frac{coefficient of x}{coefficient of x^{3} }$

Product of zeroes = $\frac{-d}{a}$ = $\frac{- constant term}{coefficient of x^{3} }$

Answer 2

$\huge\texttt{Relationship between zeroes and co - efficient of a cubic polynomial:-}$

Let ,
$p(x) = a {x}^{3} + b {x}^{2} + cx + d$, is a cubic polynomial where , $a$ is not equal to zero , and
$\alpha$, $\beta$and $\gamma$are its zeroes .

Therefore ,

$(x - \alpha )$,
$(x - \beta )$and
$(x - \gamma )$
are the factors of
$p(x)$
Now ,

$p(x) = k[{(x - \alpha )(x - \beta )(x - \gamma )}]$

Where , $k$is a constant .

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

$= k[{ {x}^{3} - {x}^{2} \gamma - {x}^{2} \alpha + x \alpha \gamma - {x}^{2} \beta + x \beta \gamma + x \alpha \beta - \alpha \beta \gamma }]$

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

Therefore ,

$a {x}^{3} +b {x}^{2} + cx + d = k {x}^{3} - {x}^{2}k( \alpha + \beta + \gamma ) + kx( \alpha \beta + \beta \gamma + \gamma \alpha ) - k \alpha \beta \gamma$

Now , By comparing the coefficients of
${x}^{3}$
${x}^{2}$
$x$
and constant , we get :-

$a = k$

$b = - k( \alpha + \beta + \gamma )$
$= > \frac{b}{ - k} = \alpha + \beta + \gamma$

$= > \frac{ - b}{a} = \alpha + \beta + \gamma$


$c = k( \alpha \beta + \beta \gamma + \gamma \alpha )$

$= > \frac{c}{k} = \alpha \beta + \beta \gamma + \gamma \alpha$

$= > \frac{c}{a} = \alpha \beta + \beta \gamma + \gamma \alpha$

And ,

$d = - k \alpha \beta \gamma$

$= > \frac{d}{ - k} = \alpha \beta \gamma$

$= > \frac{ - d}{a} = \alpha \beta \gamma$


Now , we can conclude that :

◻ The sum of the zeroes of a cubic polynomial = $\frac{-b}{a}$


◻ The product of the zeroes of a cubic polynomial when taken twice = $\alpha\beta+\beta\gamma+\gamma\alpha$


◻ The product of the zeroes of a cubic polynomial = $\alpha\beta\gamma$