Write the binomial expansion of (x^2+2/x)^5.
To expand the binomial expression (x^2 + 2/x)^5, we can use the binomial theorem. The binomial theorem allows us to expand the power of a binomial expression using combinations and powers of the individual terms.
The binomial theorem states that for any positive integer n:
(x + y)^n = C(n,0) * x^n * y^0 + C(n,1) * x^(n-1) * y^1 + C(n,2) * x^(n-2) * y^2 + ... + C(n,n) * x^0 * y^n
where C(n,k) is the binomial coefficient, given by the formula:
C(n,k) = n! / (k! * (n-k)!)
In our case, the expression is (x^2 + 2/x)^5. Let's expand it using the binomial theorem:
(x^2 + 2/x)^5 = C(5,0) * x^10 * (2/x)^0 + C(5,1) * x^8 * (2/x)^1 + C(5,2) * x^6 * (2/x)^2 + C(5,3) * x^4 * (2/x)^3 + C(5,4) * x^2 * (2/x)^4 + C(5,5) * x^0 * (2/x)^5
Now let's simplify each term:
C(5,0) * x^10 * (2/x)^0 = 1 * x^10 * 2^0 = x^10
C(5,1) * x^8 * (2/x)^1 = 5 * x^8 * 2/x = 10x^7
C(5,2) * x^6 * (2/x)^2 = 10 * x^6 * (2/x)^2 = 10 * x^6 * (2^2 / x^2) = 40x^4
C(5,3) * x^4 * (2/x)^3 = 10 * x^4 * (2/x)^3 = 10 * x^4 * (2^3 / x^3) = 80x
C(5,4) * x^2 * (2/x)^4 = 5 * x^2 * (2/x)^4 = 5 * x^2 * (2^4 / x^4) = 80/x^2
C(5,5) * x^0 * (2/x)^5 = 1 * x^0 * (2/x)^5 = (2^5 / x^5) = 32/x^5
Now let's put it all together:
(x^2 + 2/x)^5 = x^10 + 10x^7 + 40x^4 + 80x + 80/x^2 + 32/x^5
Therefore, the binomial expansion of (x^2 + 2/x)^5 is x^10 + 10x^7 + 40x^4 + 80x + 80/x^2 + 32/x^5.