Roots Progression

Let the three real roots be $a, b$ and $c$ .

From the topic of the "sum and product of roots",

we get :

$a + b + c = 33$ ---------------------------(*)

$ab + bc + ca = 354$ --------------------(**)

$abc = k$ ---------------------------------------( * )

From equation (*):

Since $a, b$ and $c$ are in an arithmetic progression, let the common difference be $n$ .

So, $(b - n) + b + (b + n) = 33$

$3b = 33$

$b = 11$

Subsituting the value of $b$ into (**), we get:

$11(a + c) + ac = 354$

$11[(b - n) + (b + n)] + ac = 354$

$22b + ac = 354$

$242 + ac = 354$

$ac = 112$

From ( * ):

$abc = k$

$(112)(11) = k$

$\boxed{k = 1232}$