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# 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.

3 years ago

Dear Student

Given factor  x2 + 2x + k = 0

Given polynomial 2x+ x-14x+ 5x + 6

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

2x+ 4x+2kx2   ( substract)
------------------------------
- 3x+(-14 - 2k)x+ 5x
- 3x-  6x- 3kx   ( substract)
------------------------------
(- 8 - 2k) x+( 5 + 3k)x + 6
(- 8 - 2k) x+(-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.

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