2. find k so that x + 2x + k is a factor of 2x4 + x3 – 14x2 +5x +6. also find all the zeroes of the two polynomials.

Dear Student

Given factor  x2 + 2x + k = 0

Given polynomial 2x4 + x3 -14x2 + 5x + 6

                        x2 + 2x + k ) 2x4 + x3 -14x2 + 5x + 6 ( 2x2 - 3x +(- 8 - 2k)

                                           2x4 + 4x3 +2kx2   ( substract)
                                         ------------------------------
                                                - 3x3 +(-14 - 2k)x2 + 5x
                                                - 3x3 -  6x2 - 3kx   ( substract)
                                               ------------------------------
                                                     (- 8 - 2k) x2 +( 5 + 3k)x + 6
                                                     (- 8 - 2k) x2 +(-16 - 4k)x + (- 8k - 2k2)  ( substract)
                                               -----------------------------------------------------------------
                                                                          ( 21 + 7k)x + (6 + 8k + 2k2) 
∴ Remainder  ( 21 + 7k)x + (6 + 8k + 2k2)  = 0

21 + 7k = 0 ⇒ k = -3.

The factors are  x2 + 2x - 3 = 0    and      2x2 - 3x - 2 = 0

                         x2 + 3x - x - 3 = 0    and      2x2 - 4x + x - 2 = 0   

                         x( x + 3 )-1( x + 3) = 0 and 2x (x - 2) + 1(x - 2) = 0         

                        (x - 1)( x + 3) = 0 and (2x + 1)(x - 2) = 0       
 x = 1 ,3 ,-1 / 2 and 2.
The zeros are 1 ,3 ,-1 / 2 and 2.

Regards

Arun (askIITians forum expert)