expand and simplify (y^2-2)^5 (y^2+2)^5 using the binomial theorem.?

Answer 1

If this is the product of (y^2-2)^5 and (y^2+2)^5, then,
using a rule of exponents, (a^n)(b^n)=(ab)^n, gives:
(y^2-2)^5(y^2+2)^5 = ((y^2-2)(y^2+2))^5 or (y^4 - 4)^5.
Now using the binomial theorem:
(y^4)^5 +5(y^4)^4(-4) + 10(y^4)^3(-4)^2 +10(y^4)^2(-4)^3 +5(y^4)(-4)^4 + (-4)^5
And simplifying:
y^20 - 20y^16 + 160y^12 -640y^8 + 1280y^4 - 1024

Answer 2

First, break this up into 5 paired multiplications products of y^2-2 and y^2+2 to get (y^4-4)^5. Then, the binomial theorum us used to crank out the terms you wind up with. There will be 6 of them
y^20
-20 y^16+
(5*2)(4*4) y^12 = 160 y^12
-(5*2)(4^3) y^8 = 640 y^8+
(5) (4^4) y^4 = 1024 y^4
-(4^5) = 1024

Answer 3

(y^2-2)^5 (y^2+2)^5
= (y^4-4)^5