Binomial Theorem

The following formula allows one to find the powers of a binomial. It is known as the binomial theorem.

Binomial Theorem

Observe that:

The number of terms is n + 1.

The coefficients are combinatorial numbers corresponding to the nth row of Pascal's triangle.

Pascal's Triangle

In the development of the binomial, the exponents of a are decreasing, one by one, from n to zero; and the exponents of b are increasing, one by one, from zero to n, therefore, the sum of the exponents of a and b in each term is equal to n.

In the case that one of the terms of the binomial is negative, alternate the positive and negative signs.


Examples

1. Binomial Theorem Exercise

Binomial Theorem Operations

Binomial Theorem Solution

2. Binomial Theorem Exercise

Binomial Theorem Operations

Binomial Theorem Solution

3. Binomial Theorem Exercise

Binomial Theorem Operations

Binomial Theorem Solution

4.Binomial Theorem Exercise

Binomial Theorem Operations

Binomial Theorem Solution

5. Binomial Theorem Exercise

Binomial Theorem Operations

Binomial Theorem Operations

Binomial Theorem Solution

6. Binomial Theorem Exercise

Binomial Theorem Operations

Binomial Theorem Solution

Calculation of the Term which Occupies the Place k

k

k

Examples

1. Find the fifth term of the development binomio.

Binomial Theorem Solution

2.Find the fourth term of the development binomio is:

Binomial Theorem Solution

3.Find the eighth term of the development binomio

Binomial Theorem Solution

4.Find the fifth term of the development Binomial Theorem Exercise.

Binomial Theorem Solution

5.Find the independent term of the development Binomial Theorem Exercise.

Binomial Theorem Solution

The exponent of a with the independent term is 0, therefore, take only the literal part and equal it to a0.

Binomial Theorem Solution

Binomial Theorem Solution

Binomial Theorem Solution