Question

find k so that x^2+2x+k is a factor of 2x^4+x^3-14x^2+5x+6​

Answer 1

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)

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

- 3x3 - 6x2 - 3kx ( substract)

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

(- 8 - 2k) x2 +(-16 - 4k)x + (- 8k - 2k2) ( substract)

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( 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.