Find the coefficient of 𝑥4

Question

Find the coefficient of 𝑥4

in the expansion of
(1 − 2𝑥)6.
A. -320
B. -240
C. 240
D. 320

Answer 1

Using the binomial theorem, we have:

(1 - 2𝑥)6 = 1 - 6(2𝑥) + 15(2𝑥)2 - 20(2𝑥)3 + 15(2𝑥)4 - 6(2𝑥)5 + (2𝑥)6

The coefficient of 𝑥4 is the coefficient of the term 15(2𝑥)4, which is 15(16) = 240. Therefore, the answer is C.

Answer 2

To find the coefficient of 𝑥^4 in the expansion of (1 - 2𝑥)^6, we can use the binomial theorem.

The binomial theorem states that for any positive integer n:

(𝑎 + 𝑏)^𝑛 = 𝐶(𝑛, 0)𝑎^𝑛 + 𝐶(𝑛, 1)𝑎^(𝑛−1)𝑏 + 𝐶(𝑛, 2)𝑎^(𝑛−2)𝑏^2 + ... + 𝐶(𝑛, 𝑛−1)𝑎𝑏^(𝑛−1) + 𝐶(𝑛, 𝑛)𝑏^𝑛,

where 𝐶(𝑛, 𝑟) is the binomial coefficient, which is calculated as:

𝐶(𝑛, 𝑟) = 𝑛! / (𝑟! (𝑛−𝑟)!)

In this case, 𝑛 = 6 and 𝑎 = 1, 𝑏 = -2𝑥. We are interested in the term with 𝑥^4, so 𝑟 = 4.

𝐶(6, 4) = 6! / (4! (6−4)!) = 6! / (4! 2!) = (6 * 5 * 4 * 3 * 2 * 1) / (4 * 3 * 2 * 1 * 2 * 1) = (6 * 5) / (2 * 1) = 15

So the coefficient of 𝑥^4 in the expansion is 15. However, we need to consider the coefficient of 𝑥^4 multiplied by -2^4 = -16.

Therefore, the coefficient of 𝑥^4 in the expansion of (1 - 2𝑥)^6 is -16 * 15 = -240.

Therefore, the correct answer is B. -240.