Find the coefficient of

Question

Find the coefficient of

𝑥3
𝑖𝑛 𝑡ℎ𝑒 𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 1
1+𝑥
A. -1
B. -6
C. 1
D. 6

Answer 1

We can use the binomial theorem to expand the expression:

(1 + x)^1 = 1 + 1x

(1 + x)^2 = 1 + 2x + 1x^2

(1 + x)^3 = 1 + 3x + 3x^2 + 1x^3

So, the coefficient of 𝑥^3 in the expansion of (1 + 𝑥)^1 is 0, and the coefficient of 𝑥^3 in the expansion of (1 + 𝑥)^2 is also 0. The coefficient of 𝑥^3 in the expansion of (1 + 𝑥)^3 is 1. Therefore, the answer is:

C. 1

Answer 2

To find the coefficient of 𝑥^3 in the expansion of (1+𝑥)^1, we can use the binomial theorem.

The binomial theorem states that for any two numbers 𝑎 and 𝑏, and any positive integer 𝑛, the 𝑛-th term in the expansion of (𝑎+𝑏)^𝑛 is given by:

𝐶(𝑛,𝑟) × 𝑎^(𝑛−𝑟) × 𝑏^𝑟

where 𝐶(𝑛,𝑟) is the binomial coefficient, which can be calculated using the formula:

𝐶(𝑛,𝑟) = 𝑛! / (𝑟!(𝑛−𝑟)!)

In this case, 𝑛 is 1, so we are looking for the coefficient of 𝑥^3 in the expansion of (1+𝑥)^1.

Using the binomial coefficient formula, we can find 𝐶(1,3):

𝐶(1,3) = 1! / (3!(1-3)!)
= 1! / (3!(-2)!)

Since (-2)! is not defined, the binomial coefficient 𝐶(1,3) is equal to 0.

Therefore, the coefficient of 𝑥^3 in the expansion of (1+𝑥)^1 is 0.