Question

find k so that x² + 2x +k is a factor of 2x⁴ + x³ -14x² + 5x +6​

Answer 1

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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)

        2x² + 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.