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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.

2. find k so that x + 2x + k is a factor of 2x4 + x3 – 14x2 +5x +6. also find all the zeroes of the two polynomials.

Grade:10

1 Answers

Arun
25763 Points
3 years ago

Dear Student

Given factor  x2 + 2x + k = 0

Given polynomial 2x+ x-14x+ 5x + 6

                        x2 + 2x + k ) 2x+ x-14x+ 5x + 6 ( 2x2 - 3x +(- 8 - 2k)

                                           2x+ 4x+2kx2   ( substract)
                                         ------------------------------
                                                - 3x+(-14 - 2k)x+ 5x
                                                - 3x-  6x- 3kx   ( substract)
                                               ------------------------------
                                                     (- 8 - 2k) x+( 5 + 3k)x + 6
                                                     (- 8 - 2k) x+(-16 - 4k)x + (- 8k - 2k2)  ( substract)
                                               -----------------------------------------------------------------
                                                                          ( 21 + 7k)x + (6 + 8k + 2k2
∴ Remainder  ( 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.

 

Regards

Arun (askIITians forum expert)

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