P(x) is a polynomial of degree more than 2 when p(x) is divided by x-2 it leaves a remainder 1 and when it is divided by x-3 it leaves a remainder 3. Find the remainder when p)x) is divided by(x-2)(x-3) ?

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Answer 1 · 2018-06-24T17:42:32

The remainder is #=x#

By the remainder theorem

When the polynomial #P(x)# is divided by #(x-2)#, the remainder is #=2#

#=>#, #P(2)=2#

When the polynomial is divided by #(x-3)#, the remainder is #=3#

#=>#, #P(3)=3#

Let #D=(x-2)(x-3)#

When the polynomial is divided by #(x-2)(x-3)#, the quotient is #Q# and the remainder #R# will be of the form #Ax+B#

Therefore,

#P(x)=QD+Ax+B#

So,

#P(2)=Q*0+2A+B#

#=>#, #2A+B=2#...............#(1)#

#P(3)=Q*0+3A+B#

#=>#, #3A+B=3#....................#(2)#

Solving equations #(1)# and #(2)# for #A# and #B#

#{(2A+B=2),(3A+B=3):}#

#=>#, #{(2A+B=2),(A=1):}#

#=>#, #{(A=1),(B=0):}#

Therefore,

The remainder is #=x#