How do you use the remainder theorem to determine the remainder when the polynomial #(n^4-3n^2-5n+2)/(n-2)#?

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Answer 1 · 2016-12-30T14:36:15

#R=-4#

The remainder theorem states that if a polynomial #" "P(x)" "# is divided by the linear factor #(x-a)" "#then the remainder is #" "P(a)#

proof;

let #" "P(x) " "#be divided by #" "(x-a)" "# to give quotient #" "Q(x)" "#and remainder #" "R#

then:#" "P(x)=(x-a)Q(x)+R#

so:#" "P(a)=cancel((a-a)Q(a))+R#

#" ":.P(a)=R" "#as required.

In this case we have :#(n^4-3n^2-5n+2)-:(n-2)#

#P(n)=n^4-3n^2-5n+2#

the divisor is #" "n-2=>a=2#

#R=P(2)=2^4-3xx2^2-5xx2+2#

#R=P(2)=16-12-10+2=-4#