Question

If $\alpha ,\beta \text{ and }\gamma$ are the roots of the equation $x^3+p{x}^2+qx+r$=0, then the coefficient of x in the cubic equation whose roots are $\alpha (\beta +\gamma ),\beta (\gamma +\alpha )\text{ and }\gamma (\alpha +\beta )$ is?

(a) 2q
(b) $q^2+pr$
(c) $p^2-qr$
(d) r(pq-r)

Answer 1

Hint: To solve this problem, we make use of the basic properties of a cubic polynomial related to the relation of sum of roots, product of roots and product of roots taken two at a time. That is,

Complete step-by-step answer:
For, $x^3+p{x}^2+qx+r$=0,

$\alpha +\beta +\gamma$=-p

$\alpha \beta +\beta \gamma +\gamma \alpha$=q

$\alpha \beta \gamma$=-r

We make use of these properties to find the cubic equation with new roots.

We have the roots as $\alpha (\beta +\gamma ),\beta (\gamma +\alpha )\text{ and }\gamma (\alpha +\beta )$. We know how to find the sum of roots, product of roots and product of roots taken two at a time in case of these new roots.

In the question in particular, we need to find the coefficient of x for the cubic polynomial-
$x^3+p{x}^2+qx+r$=0

Coefficient of x is given by product of roots of the cubic equation taken two at a time. In the normal cubic polynomial, this was $\alpha \beta +\beta \gamma +\gamma \alpha$.

In case of the new roots, $\alpha (\beta +\gamma ),\beta (\gamma +\alpha )\text{ and }\gamma (\alpha +\beta )$, the product of the new roots taken two at a time is –
=$\alpha \beta (\beta +\gamma )(\gamma +\alpha )\text{ + }\beta \gamma (\gamma +\alpha )(\alpha +\beta )+\gamma \alpha (\alpha +\beta )(\beta +\gamma )$
We now expand each of these three terms, we get,

=$[{\alpha }^2{\beta }^2\text{+}{\alpha }^2\beta \gamma +\alpha {\beta }^2\gamma +\alpha \beta {\gamma }^2+{\beta }^2{\gamma }^2+{\alpha }^2\beta \gamma +\alpha {\beta }^2\gamma +\alpha \beta {\gamma }^2+{\gamma }^2{\alpha }^2+{\alpha }^2\beta \gamma +\alpha {\beta }^2\gamma +\alpha \beta {\gamma }^2$]

=$${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2\text{+3}{\alpha }^2\beta \gamma +3\alpha {\beta }^2\gamma +3\alpha \beta {\gamma }^2$$

=$${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2\text{+3}\alpha \beta \gamma (\alpha +\beta +\gamma )$$-- (1)

Now, we know that,

q=$\alpha \beta +\beta \gamma +\gamma \alpha$

Squaring LHS and RHS, we get,

$q^2=$${(\alpha \beta +\beta \gamma +\gamma \alpha )}^2$

We use the following algebraic identity,

${(a+b+c)}^2=a^2+b^2+c^2+2(ab+bc+ca)$
Thus, we have,

$q^2=$$${(\alpha \beta )}^2+{(\beta \gamma )}^2+{(\gamma \alpha )}^2+2(\alpha \beta )(\beta \gamma )+2(\beta \gamma )(\gamma \alpha )+2(\gamma \alpha )(\alpha \beta )$$

Thus, be re-arranging, we would have,

$q^2=$$${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2\text{+2}\alpha \beta \gamma (\alpha +\beta +\gamma )$$-- (2)

Now, from (1)

$\alpha \beta (\beta +\gamma )(\gamma +\alpha )\text{ + }\beta \gamma (\gamma +\alpha )(\alpha +\beta )+\gamma \alpha (\alpha +\beta )(\beta +\gamma )$=$${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2\text{+3}\alpha \beta \gamma (\alpha +\beta +\gamma )$$

Thus,

$${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2\text{+3}\alpha \beta \gamma (\alpha +\beta +\gamma )$$=$${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2\text{+2}\alpha \beta \gamma (\alpha +\beta +\gamma )+\alpha \beta \gamma (\alpha +\beta +\gamma )$$
From (2), we have,

$${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2\text{+3}\alpha \beta \gamma (\alpha \beta +\beta \gamma +\gamma \alpha )$$= $q^2+(-p)(-r)$

Since, we know that,

$\alpha +\beta +\gamma$=-p

$\alpha \beta \gamma$=-r

Thus, we have,

$${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2\text{+3}\alpha \beta \gamma (\alpha \beta +\beta \gamma +\gamma \alpha )$$=$q^2+pr$

Hence, the correct answer is (b) $q^2+pr$.

Note: For solving problems related to roots of a cubic polynomial, we should know the basic properties of sum of roots, product of roots and product of roots taken two at a time. Further, it is then to solve the problem, one should be aware about the basic manipulations involving algebraic terms like re-grouping, re-arrangement and usage of the known properties.