Expansion of (1 x)ki power n

The expansion of (1 x)ki power n can be computed using the binomial theorem. The binomial theorem is a formula used to expand the powers of binomials, which are expressions with two terms connected by addition or subtraction. In this case, the binomial is (1 x) and we are raising it to the power of n.

The binomial theorem states that the expansion of (1 x)ki power n is given by the sum of the binomial coefficients multiplied by the respective powers of 1 and x. The binomial coefficients can be calculated using the formula nCi = n! / (i!(n-i)!), where n is the power and i is the term number.

Let's break down the expansion of (1 x)ki power n step by step:

1. Determine the number of terms in the expansion:
The number of terms in the expansion is (n + 1), as the powers of x range from 0 to n.

2. Calculate the binomial coefficients:
For each term i, calculate the binomial coefficient nCi using the formula mentioned above.

3. Expand the powers of 1 and x:
For each term i, multiply the respective powers of 1 and x. The power of 1 is (n - i) and the power of x is i.

4. Write out the expanded form:
Write out each term individually, combining the binomial coefficients, powers of 1, and powers of x.

The expanded form of (1 x)ki power n will be in the form of a polynomial with n+1 terms.

Let's illustrate this with an example:

Suppose we want to expand (1 + x)³.

1. Determine the number of terms:
The number of terms is (3 + 1) = 4.

2. Calculate the binomial coefficients:
nC0 = 3! / (0!(3-0)!) = 1
nC1 = 3! / (1!(3-1)!) = 3
nC2 = 3! / (2!(3-2)!) = 3
nC3 = 3! / (3!(3-3)!) = 1

3. Expand the powers of 1 and x:
(1)³ = 1
(x)⁰ = 1
(x)¹ = x
(x)² = x²
(x)³ = x³

4. Write out the expanded form:
The expanded form is: 1 + 3x + 3x² + x³

Thus, the expansion of (1 + x)³ is 1 + 3x + 3x² + x³.

By following the steps of the binomial theorem, you can expand any (1 x)ki power n expression.

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