Question

Factorize :- 5x 3 - 24x 2 + 55x + 6

Answer 1

5 x³ - 24 x² + 55 x + 6

(x - α) (5 x² + a x + 6/α)
One root  α = - 0.10424586

-5 α + a = -24       so   a =  -24.521229
0.104 a - 6/α = 55

to factorize   5 x² - (24 - 5 α) x + 6/α 
     roots  = 1.734063389    and   -6.638309247
so factors are :  (x + 0.10424586)(x - 1.734063389) (x + 6.638209247)
======================
Roots of  P(x) = 5 [ x³ - 4.8 x² + 11 x + 1.2 ] = 0

let x = (y + 1.6)
so P(x) /5 =
  y³+4.096+4.8 y² +7.68 y -4.8 (y² +2.56 +3.2 y)+11y+17.6+ 1.2= 0

so  P(y) :  y³ + 3.32 y + 10.608  = 0

      Now let  y = t - 3.32 /(3t)

P(y): t³- 3.32³ /27t³ - 3*3.32 t/3 + 3*3.32² /9t + 3.32 t -3.32² /3t +10.608= 0
         :   t³ - (3.32/3)³ /t³ + 10.608 = 0
             t⁶ + 10.608 t³ - (3.32/3)³ = 0
    so t³ = roots =  [5.304 +- √(5.304² + (3.32/3)³ ) ] 

         now find  t³  and then  t .
          then find    y 
         then find x   roots of  P(x).

then express the factors: