How do you evaluate #(a ^ { n + 2} - 2a ^ { n } + 3a ^ { n + 1} ) ( a ^ { n } + a ^ { n + 1} )#?

1 Answer
Tamir E.
Nov 19, 2017

#-2a^(2n)+a^(2n+1)+4a^(2n+2)+a^(2n+3)#
OR
#a^(2n)*(-2+a+4a^2+a^3)#

Explanation:

Laws:
1) #x^a*x^b=x^(a+b)#
2) #b*x^a+c*x^a=(b+c)*x^(a)#

#(a^(n+2)-2a^n+3a^(n+1))(a^n+a^(n+1))=#
#=a^(n+2+n)+a^(n+2+n+1)-2a^(n+n)-2a^(n+n+1)+3a^(n+n+1)+3a^(n+1+n+1)=#
#=a^(2n+2)+a^(2n+3)-2a^(2n)-2a^(2n+1)+3a^(2n+1)+3a^(2n+2)=#
#=4a^(2n+2)+a^(2n+3)-2a^(2n)+a^(2n+1)=#
#=-2a^(2n)+a^(2n+1)+4a^(2n+2)+a^(2n+3)=#

#=a^(2n)*(-2+a+4a^2+a^3)#

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