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

1 Answer
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)