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  4. If the roots of x3-42 x2+336 x-512=0, are in increasing geometric progression, then its common ratio is

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Q. If the roots of $x^{3}-42 x^{2}+336 x-512=0$, are in increasing geometric progression, then its common ratio is

EAMCETEAMCET 2013

Solution:

Given, cubic equation is
$x^{3}-42 x^{2}+336 x-512=0$
$\Rightarrow x^{2}(x-2)-40 x(x- 2)+256(x-2)=0$
$\Rightarrow (x-2)\left(x^{2}-40 x+256\right)=0$
$\Rightarrow (x-2)\left\{x^{2}-32 x-8 x+256\right\}=0$
$\Rightarrow (x-2)\{x(x-32)-8(x-32)\}=0$
$\Rightarrow (x-2)(x-32)(x-8)=0$
$\Rightarrow (x-2)(x-8)(x-32)=0$
$\Rightarrow x=2,8,32$
Which represents a geometric progression in increasing order.
Common ratio $=\frac{8}{2}=4: 1$