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2. find k so that x + 2x + k is a factor of 2x 4 + x 3 – 14x 2 +5x +6. also find all the zeroes of the two polynomials.
Dear StudentGiven factor x2 + 2x + k = 0Given polynomial 2x4 + x3 -14x2 + 5x + 6 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) ∴ Remainder ( 21 + 7k)x + (6 + 8k + 2k2) = 021 + 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. RegardsArun (askIITians forum expert)
Dear Student
Given factor x2 + 2x + k = 0
Given polynomial 2x4 + x3 -14x2 + 5x + 6 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) ∴ Remainder ( 21 + 7k)x + (6 + 8k + 2k2) = 021 + 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.
Regards
Arun (askIITians forum expert)
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