How do you use the remainder theorem to determine the remainder when the polynomial #(2t-4t^3-3t^2)/(t-2)#?

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Answer 1 · 2017-03-31T06:30:57

The remainder is #-40#

According to remainder theorem, when a polynomial function #f(x)# is divided by #(x-a)#, the remainder is #f(a)#.

Here we have the function #f(t)=-4t^3-3t^2+2t#, which is divided by #t-2# and hence the remainder is

#f(2)=-4xx2^3-3xx2^2+2xx2#

#=-4xx8-3xx4+4#

#=-32-12+4=-40#

One can check it too

#f(t)=-4t^3-3t^2+2t#

= #-4t^2(t-2)-8t^2-3t^2+2t=-4t^2(t-2)-11t^2+2t#

(we have subtracted #8t^2# to compensate for #-4t^2xx(-2)=8t^2#)

= #-4t^2(t-2)-11t(t-2)-22t+2t=-4t^2(t-2)-11t(t-2)-20t#

= #-4t^2(t-2)-11t(t-2)-20(t-2)-40#

= #(-4t^2-11t-20)(t-2)-40#

i.e. quotient is #(-4t^2-11t-20)# and remainder is #-40#