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CBSE Class 10 Answered

Please help me to solve this question  When x2  - 2x + k divides the polynomial x4 - 6x3 + 16x2 - 25x + 10, the remainder is (x + a) . The value of a is _______
Asked by ratendra.johri | 28 May, 2018, 08:00: PM
answered-by-expert Expert Answer
begin mathsize 16px style straight P left parenthesis straight x right parenthesis space equals space straight x to the power of 4 space minus space 6 straight x cubed space plus space 16 space straight x to the power of 2 space end exponent minus 25 straight x space plus space 10 space is space divided space by space straight x squared space minus 2 straight x space plus straight k.
Let space the space quotient space be space straight x squared thin space plus space bx space plus space straight c
Remainder equals space straight x space plus space straight a space
straight P left parenthesis straight x right parenthesis space equals space divisor space cross times quotient space plus space remainder
rightwards double arrow straight P left parenthesis space straight x right parenthesis space equals left parenthesis space straight x squared space minus 2 straight x space plus straight k right parenthesis left parenthesis space straight x squared thin space plus space bx space plus space straight c right parenthesis space plus space left parenthesis straight x space plus space straight a space right parenthesis space equals space straight x to the power of 4 space plus space straight x cubed left parenthesis straight b minus 2 right parenthesis plus straight x squared left parenthesis space straight c minus space 2 straight b space plus space straight k right parenthesis space plus straight x space left parenthesis bk space minus 2 straight c space plus 1 right parenthesis space plus space ck space plus straight a
On space space Comparing space straight P left parenthesis straight x right parenthesis comma space we space get space
rightwards double arrow straight b space minus space 2 equals space minus 6 space rightwards double arrow straight b space equals space minus 4.... left parenthesis 1 right parenthesis
left parenthesis space straight c minus space 2 straight b space plus space straight k right parenthesis space equals space 16
rightwards double arrow straight c space plus space straight k space equals space 8 space space....... left parenthesis 2 right parenthesis
bk space minus 2 straight c space plus space 1 space equals negative 25
2 straight k space plus space straight c space equals space 13 space..... left parenthesis 3 right parenthesis
After space soliving space left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma space we space get
straight k space equals space 5 space and space straight c space equals space 3
equating space the space constant space terms comma space we space get
ck space plus straight a space equals space 10
rightwards double arrow straight a space equals space minus 5.

end style
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