Expand the following: [(a+b^2)(a-b^2)]^5Show the complete solution to explain the answer.

Well, this polynomial first has a difference of squares
inside

Mutiplying them
out

(a+b^2)(a-b^2)=a^2-b^4

The
whole polynomial
becomes

(a^2-b^4)^5

Using
Pascal's Triangle and the Binomial Theorem:

1              
       first degree

1  2  1                  second
degree

1   3   3   1               third
degree

1  4   6  4   1             fourth
degree

1 5  10 10 5 1           fifth
degree

so the polynomial (a^2-b^4))^5= (a^2))^5-
5((a^2)^4))b^4 + 10
((a^2)^3)((b^4)^2)-10((a^2)^2)((b^4)^3)+5a^2((b^4)^4)+(b^4)^5

simplify
this huge expression

=
a^10-5(a^8)*(b^4)+10(a^6)*(b^8)-10(a^4)*(b^12)+5(a^2)*(b^16)-b^20

Actually,
for any polynomial to the exponent
5

(x+y)^5=x^5+5(x^4)*y+10(x^3)*(y^2)+10(x^2)*(y^3)+

5x*(y^4)+y^5

You
could see, the x-exponent is decreasing by one every term, and the y exponent is
increasing by one in every term.

The
coefficents are given by the Pascal Triangle.