CBSE Class 9 Answered

If (x - 2) is a factor of the polynomial x³ + ax² + bx + 6 and when divided by (x - 3), it leaves remainder 3. Find the values of a and b.

Asked by Topperlearning User | 14 Sep, 2017, 01:52: PM

Expert Answer

$begin mathsize 12px style Let space straight p left parenthesis straight x right parenthesis space equals space straight x cubed space plus space ax squared space plus space bx space plus space 6 left parenthesis straight x space minus space 2 right parenthesis space is space straight a space factor space of space the space polynomial space straight x cubed space plus space ax squared space plus space bx space plus 6 straight p left parenthesis 2 right parenthesis space equals space 0 straight p left parenthesis 2 right parenthesis space equals space 2 cubed space plus space straight a space cross times space 2 squared space plus space straight b space cross times space 2 space plus space 6 space space space space space space space space space equals space 8 space plus space 4 straight a space plus space 2 straight b space plus space 6 space space space space space space space space space space equals 14 space plus space 4 straight a space plus space 2 straight b space space space space space space space space space space equals space 0 7 space plus space 4 straight a space plus space straight b space equals 0 straight b space equals space minus 7 space minus space 4 straight a space.......... left parenthesis straight i right parenthesis space straight x cubed space plus space ax squared space plus space bx space plus space 6 space when space divided space by space left parenthesis straight x space minus space 3 right parenthesis space leaves space remainder space 3. straight p left parenthesis 3 right parenthesis space equals space 3 straight p left parenthesis 3 right parenthesis space equals space 3 cubed space plus space straight a space cross times space 3 squared space plus space straight b space cross times space 3 space plus space 6 space space space space space space space space space space equals space 27 space plus space 9 straight a space plus space 3 straight b space plus space 6 space space space space space space space space space space space space equals space 33 space plus space 9 straight a space plus space 3 straight b space space space space space space space space space space space space equals space 3 11 space plus space 3 straight a space plus space straight b space equals space 1 3 straight a space plus space straight b space equals space minus 10 straight b space equals space minus 10 space minus space 3 straight a space.......... left parenthesis ii right parenthesis Equating space the space value space of space straight b space from space left parenthesis ii right parenthesis space and space left parenthesis straight i right parenthesis space we space have minus 7 space minus space 2 straight a space equals space minus space 10 space minus space 3 straight a straight a space equals space minus 3 Substituting space straight a space equals space minus 3 space in space left parenthesis straight i right parenthesis space we space get straight b space equals space minus 7 space minus space 2 space left parenthesis space minus 3 right parenthesis space equals space minus 7 space plus space 6 space equals space minus 1 Thus space the space values space of space straight a space and space straight b space are space minus 3 space and space minus 1 space respectively. end style$