Find the value of the coefficient of x^4 in the expansion of (x - 2)^6.

Find the value of the coefficient of x^4 in the expansion of (x - 2)^6.

Step 1: Identify the expression we need to expand, which is (x - 2)^6.
Step 2: Recognize that we will use the binomial theorem to find the coefficient of x^4.
Step 3: In the binomial theorem, the general term in the expansion of (a + b)^n is given by nCk * a^(n-k) * b^k.
Step 4: Here, n = 6, a = x, and b = -2.
Step 5: We want the coefficient of x^4, which means we need to find the term where the power of x is 4.
Step 6: To find this term, we set n - k = 4, which means k = 6 - 4 = 2.
Step 7: Now we can calculate the coefficient using k = 2: it will be 6C2 * (x)^(6-2) * (-2)^2.
Step 8: Calculate 6C2, which is the number of ways to choose 2 items from 6, and it equals 15.
Step 9: Calculate (x)^(6-2) = (x)^4, which is just x^4.
Step 10: Calculate (-2)^2, which equals 4.
Step 11: Now combine these results: the coefficient is 15 * 1 * 4 = 60.

Binomial Theorem – The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where the coefficients can be calculated using combinations.
Coefficient Extraction – Identifying the specific coefficient of a term in a polynomial expansion, particularly focusing on the powers of x.