Answer:
The coefficient of x² in the expansion of (x-2/x)⁶ is 48.
Explanation:
To find the coefficient of x² in the expansion of (x-2/x)⁶, we need to apply the binomial theorem.
The binomial theorem states that:
(a + b)ⁿ = Σ (n choose k) a^(n-k) b^k,
where Σ is the sum over all k from 0 to n, and (n choose k) is the binomial coefficient, given by:
(n choose k) = n! / (k! (n-k)!).
In our case, we have:
a = x, b = -2/x, and n = 6.
So, using the binomial theorem, we can expand (x-2/x)⁶ as:
(x - 2/x)⁶ = Σ (6 choose k) x^(6-k) (-2/x)^k
Now we want to find the coefficient of x², which means we need to find the term with x^(6-2) = x^4. To get an x^4 term, we need to choose k = 2 in the sum above:
(x - 2/x)⁶ = (6 choose 0) x^6 (-2/x)^0 + (6 choose 1) x^5 (-2/x)^1 + (6 choose 2) x^4 (-2/x)^2 + ...
= x^6 - 12x^4 + 48x² - 64/ x² + ...
So the coefficient of x² in the expansion of (x-2/x)⁶ is 48.