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Find k so that x + 2x + k is a factor of 2x4 + x - 14x + 5x + 6. Also find all thezeroes of the two polynomials. |
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Answer» 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. 1;2;3;-1/2 are zeros of given polynomial. |
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