25. find k so that x + 2x + k is a factor of 2x + x - 14x2 + 5x + 6. a
polynomials.

Hi!

Here is my solution:

Given factor: x2 + 2x + k = 0

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

Divide the polynomial by the factor

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)

The remainder is: ( 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.

Step-by-step explanation: