Finding the Constant Term in the Binomial Expansion
To find the constant term in the binomial expansion of (2x - (5/x^3))^12, we need to determine the term where the power of 'x' is zero.
Step-by-Step Explanation:
- Use the general term of the binomial expansion:
T_(k+1) = C(12, k) * (2x)^(12-k) * (-(5/x^3))^k - Simplify the expression:
T_(k+1) = C(12, k) * 2^(12-k) * x^(12-k) * (-5)^k * x^(-3k) - Combine the powers of 'x':
T_(k+1) = C(12, k) * 2^(12-k) * (-5)^k * x^(12-4k) - Set the exponent of 'x' to zero:
12 - 4k = 0 - Solve for 'k':
k = 3 - Substitute 'k = 3' back into the expression for T_(k+1):
T_4 = C(12, 3) * 2^(12-3) * (-5)^3 * x^0 - Calculate the constant term:
C(12, 3) = 220
2^9 = 512
(-5)^3 = -125 - Multiply these values to get the constant term:
220 * 512 * (-125) = -14080000
Hence, the constant term in the expansion of (2x - (5/x^3))^12 is -14080000.
