The Binomial Theorem

Patterns in Binomial Expansions By studying the expanded form of each binomial expression, we are able to discover the following patterns in the resulting polynomials. 1. The first term is an. The exponent on a decreases by 1 in each successive term. 2. The exponents on b increase by 1 in each successive term. In the first term, the exponent on b is 0. (Because b0 = 1, b is not shown in the first term.) The last term is bn. 3. The sum of the exponents on the variables in any term is equal to n, the exponent on (a + b)n. 4. There is one more term in the polynomial expansion than there is in the power of the binomial, n. There are n + 1 terms in the expanded form of (a + b)n. Using these observations, the variable parts of the expansion (a + b)6 are a6, a5b, a4b2, a3b3, a2b4, ab5, b6.

Patterns in Binomial Expansions Let's now establish a pattern for the coefficients of the terms in the binomial expansion. Notice that each row in the figure begins and ends with 1. Any other number in the row can be obtained by adding the two numbers immediately above it. Coefficients for (a + b)1. Coefficients for (a + b)2. Coefficients for (a + b)3. Coefficients for (a + b)4. Coefficients for (a + b)5. Coefficients for (a + b)6. 1 2 1 3 3 1 4 6 4 1 5 10 10 5 1 1 6 15 20 15 6 1 The above triangular array of coefficients is called Pascal’s triangle. We can use the numbers in the sixth row and the variable parts we found to write the expansion for (a + b)6. It is (a + b)6 = a6 + 6a5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab5 + b6

Definition of a Binomial Coefficient . For nonnegative integers n and r, with n > r, the expression is called a binomial coefficient and is defined by

Example Evaluate Solution:

A Formula for Expanding Binomials: The Binomial Theorem For any positive integer n,

Example Expand Solution:

Example cont. Expand Solution:

Finding a Particular Term in a Binomial Expansion The rth term of the expansion of (a+b)n is

Example Find the third term in the expansion of (4x-2y)8 Solution: (4x-2y)8 n=8, r=3, a=4x, b=-2y

Text Example Find the fourth term in the expansion of (3x + 2y)7. Solution We will use the formula for the rth term of the expansion (a + b)n, to find the fourth term of (3x + 2y)7. For the fourth term of (3x + 2y)7, n = 7, r = 4, a = 3x, and b = 2y. Thus, the fourth term is

The Binomial Theorem