Question

45. Find k so that x(power 2)+ 2x + k is a factor of 2x(power 4)+X(power 3)-14X(power 2)+5x+6. Also find all the zeroes of the two
polynomials.​

Answer 1

1. Question⇒    Find k so that x(power 2)+ 2x + k is a factor of 2x(power 4)+X(power 3)-14X(power 2)+5x+6. Also find all the zeroes of the two   polynomials.​

Step-by-step explanation:

• 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.