,a) Find k so that x +2x+k is a factorof 2x4+x-14x2+5x+6. Also find al

1.	,a) Find k so that x +2x+k is a factorof 2x4+x-14x2+5x+6. Also find all the zeroes ofthe two polynomial.
Answer» 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.

,a) Find k so that x +2x+k is a factorof 2x4+x-14x2+5x+6. Also find all the zeroes ofthe two polynomial.

Answer»

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.