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Question :-

If the polynomial x4−6x3+16x2−25x+10 is divided by another polynomial x2−2x+k, the remainder comesout to be x+a, find k and a.

Answer:-

We know that           Dividend = Divisor × Quotient + Remainder or Dividend−Remainder = Divisor × Quotient or (x4−6x3+16x2−25x+10)−(x+a)=(x2−2x+k)×Quotient or x4−6x3+16x2−26x+10−a=(x2−2x+k)×QuotientNow,x2−2x+kx2−4x+(8−k)x4−6x3+16x2−26x+10−a                                                         x4−2x3+      kx2−    +          −                                                      ¯                              −4x3+(16−k)x2−26x+10−a       −4x3+    8x2           −  4kx       +         −                        +¯                                         −(8−k)x2+(4k−26)x+10−a                   −(8−k)x2−(−2k+16)x+8k−k2                   +                     +                            −       +                 ¯                                           (2k−10)x+k2−8k+10−aRemainder  (2k−10)x+k2−8k+10−a must be zero. So,      (2k−10)x+k2−8k+10−a =0⇒2k−10=0 or k2−8k+10−a=0⇒k=5  or 52−8×5+10−a=0⇒k=5  and  a=−5

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