Find the coefficient of the x^2 term in the expansion of (3x - (2/x^2))^8
I would advice to take out the 1/x^2 term first:
(3x - (2/x^2))^8 =
1/x^16 (3 x^3 - 2)^8 =
So you need to get x^18 from the expansion of the bracket:
(3 x^3 - 2)^8 contains a term:
(3 x^3)^6 (-2)^2 Binomial(8,6)
So, the coefficient is 81648.
To find the coefficient of the x^2 term in the expansion of (3x - (2/x^2))^8, we can use the binomial theorem.
The binomial theorem states that the expansion of (a + b)^n can be found using the formula:
(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, r) * a^(n-r) * b^r + ... + C(n, n) * a^0 * b^n
where C(n, r) represents the number of combinations of choosing r elements from a set of n elements, and can be calculated using the formula:
C(n, r) = n! / (r! * (n-r)!)
In this case, a = 3x and b = - (2/x^2), and we are interested in finding the coefficient of the x^2 term, which corresponds to the term with r = 2.
Using the binomial theorem, the coefficient of the x^2 term is given by:
C(8, 2) * (3x)^(8-2) * (- (2/x^2))^2
We can simplify this expression by calculating the combination C(8, 2) and simplifying the powers of x and the constant term:
C(8, 2) = 8! / (2! * (8-2)!) = 8! / (2! * 6!) = (8 * 7) / (2 * 1) = 28
(3x)^(8-2) = (3x)^6 = 3^6 * x^6 = 729x^6
(- (2/x^2))^2 = (-2/x^2)^2 = (-2)^2 / (x^2)^2 = 4 / x^4
Therefore, the coefficient of the x^2 term is given by:
C(8, 2) * (3x)^(8-2) * (- (2/x^2))^2 = 28 * 729x^6 * 4 / x^4 = 81,792x^6 / x^4 = 81,792x^2
So, the coefficient of the x^2 term in the expansion of (3x - (2/x^2))^8 is 81,792.