Solve given the root x^6 + 2x^5 - 13x^4 -10x^3 + 40x^2 = 0; x-2

Question

 Solve the polynomial equation x⁶ +2x⁵ - 13x⁴ -10x³ + 40x²= 0 ; given x - 2 is a root.

Answer 1

X⁶ + 2X⁵ - 13X⁴ - 10X³ + 40X²  = 0

 

X - 2 is a factor, 2 is a root

 

synthetically divide by 2

 

        1     2     -13    -10       40     0   

 

2            2         8     -10     -40    0    

 

       1     4       -5      -20       0     0    

 

Therefor, the original polynomial factors into:

 

  ( X - 2 ) ( X⁵ + 4X⁴ -5X³ - 20X² ) = 0

 

factor X² out of second parenthesis:

 

   X² ( X - 2)( X³ + 4x² - 5X - 20 ) = 0

 

  factor last parenthesis

 

   X² ( X - 2 )[ X² ( X + 4 ) - 5 ( X + 4 )] = 0

 

   X² ( X - 2 ) ( X²  - 5 ) ( X + 4 ) = 0

 

Set all factors equal to 0 to find all zeros

 

X² = 0   ;             X - 2 = 0   ;      X² - 5 = 0   ;       X + 4 = 0

 

X = 0    ;                  X = 2   ;         X² = 5  ;              X = -4

multiplicity

of 2, graph                                      X = ±√5

is tangent

at (0,0)