Let #f(x)=(x-2)^2013-x^2014#, find the remainder when #[f(x+1)]^2# is divided by x?

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Answer 1 · 2018-06-25T14:29:04

# 4#.

#f(x)=(x-2)^2013-x^2014#.

# :. f(x+1)={(x+1)-2}^2013-(x+1)^2014,#

#=(x-1)^2013-(x+1)^2014#.

# rArr [f(x+1)]^2=[(x-1)^2013-(x+1)^2014]^2#.

Let, #p(x)=[f(x+1)]^2=[(x-1)^2013-(x+1)^2014]^2#.

By the Remainder Theorem, we know that, when #p(x)# is

divided by #(x-alpha)#, the remainder is #p(alpha)#.

#"Consequently, "p(x)," when divided by "x=(x-0),"#

#"will leave the remainder "p(0)=[(-1)^2013-(1)^2014]^2=4#.