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If the polynomial x4−6x3+16x2−25x+10 is dividedby another polynomial x2−2x+k, the remainder comesout to be x+a, find k and a.
We know that Dividend = Divisor × Quotient + Remainderor Dividend−Remainder = Divisor × Quotientor (x4−6x3+16x2−25x+10)−(x+a)=(x2−2x+k)×Quotientor 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
