Question

If $\alpha ,\beta$ are roots of $a{x}^2-2bx+c=0$ then $a^3{\beta }^3+a^2{\beta }^2+a^3{\beta }^2$ is

• A. $\frac{c^2(c+2b)}{a^3}$
• B. $\frac{b{c}^2}{a^3}$
• C. $\frac{c^2}{a^3}$
• D. None

Answer 1

The correct option is A $\frac{c^2(c+2b)}{a^3}$

The given expression $\Rightarrow {\alpha }^3{\beta }^3+{\alpha }^2{\beta }^3+{\alpha }^3{\beta }^2$ can be written as $\Rightarrow (\alpha \beta {)}^3+(\alpha \beta {)}^2(\alpha +\beta )$ ....$\left(1\right)$

Now as we know that $\alpha$ and $\beta$ are the roots of the quadrtic equation $a{x}^2-bx+c=0$

So sum of the roots $\alpha +\beta =-\frac{-2b}{a}=\frac{2b}{a}$

Also the product of the roots $\alpha .\beta =\frac{c}{a}$

By putting the value of the terms $\alpha +\beta$ and $\alpha .\beta$ into equation $\left(1\right)$

We get, $\Rightarrow (\alpha \beta {)}^3+(\alpha \beta {)}^2(\alpha +\beta )$ $=(\frac{c}{a}{)}^3+(\frac{c}{a}{)}^2\left(\frac{2b}{a}\right)$

$\Rightarrow (\alpha \beta {)}^3+(\alpha \beta {)}^2(\alpha +\beta )$ $=\frac{c^2(c+2b)}{a^3}$

Hence the Correct option is $A$