Find the value of the coefficient of x^4 in the expansion of (x - 2)^6.
Practice Questions
1 question
Q1
Find the value of the coefficient of x^4 in the expansion of (x - 2)^6.
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Using the binomial theorem, the coefficient of x^4 in (a + b)^n is given by nCk * a^(n-k) * b^k. Here, n=6, a=x, b=-2, and k=2. Thus, the coefficient is 6C2 * (1)^4 * (-2)^2 = 15 * 4 = 60.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the value of the coefficient of x^4 in the expansion of (x - 2)^6.
Solution: Using the binomial theorem, the coefficient of x^4 in (a + b)^n is given by nCk * a^(n-k) * b^k. Here, n=6, a=x, b=-2, and k=2. Thus, the coefficient is 6C2 * (1)^4 * (-2)^2 = 15 * 4 = 60.
Steps: 11
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.
