Pascal’s Triangle With the coefficients arranged in this way, each number in the triangle is the sum of the two numbers directly above it (one to the right and one to the left). For example, in row four, 1 is the sum of 1 (the only number above it), 4 is the sum of 1 and 3, 6 is the sum of 3 and 3, and so on. This triangular array of numbers is called Pascal’s triangle, in honor of the seventeenth-century mathematician Blaise Pascal. It was, however, known long before his time.
Pascal’s Triangle To find the coefficients for (x + y) 6 we need to include row six in Pascal’s triangle. Adding adjacent numbers, we find that row six is Using these coefficients, we obtain the expansion of (x + y) 6 :
n-Factorial Although it is possible to use Pascal’s triangle to find the coefficients of (x + y) n for any positive integer n, this calculation becomes impractical for large values of n because of the need to write all the preceding rows. A more efficient way of finding these coefficients uses factorial notation. The number n! (read “n-factorial”) is defined as shown in the next slide.
n-Factorial For any positive integer n,
Binomial Coefficients Now look at the coefficients of the expansion The coefficient of the second term, 5x 4 y, is 5, and the exponents on the variables are 4 and 1. Note that
Binomial Coefficient For nonnegative integers n and r, with r ≤ n,
Binomial Coefficients The binomial coefficients are numbers from Pascal’s triangle. For example, is the first number in row three, and is the fifth number in row seven. Graphing calculators are capable of finding binomial coefficients. A calculator with a 10-digit display will give exact values for n! for n ≤ 13 and approximate values of n! for 14 ≤ n ≤ 69.
Example 1 EVALUATING BINOMIAL COEFFICIENTS Solution d. Evaluate the binomial coefficient.
The Binomial Theorem Our observations about the expansion of (x + y) n are summarized as follows. 1. There are n + 1 terms in the expansion. 2. The first term is x n, and the last term is y n. 3. In each succeeding term, the exponent on x decreases by 1 and the exponent on y increases by The sum of the exponents on x and y in any term is n. 5. The coefficient of the term with x r y n – r or x n – r y r is
Binomial Theorem For any positive integer n and any complex numbers x and y,
Note The binomial theorem looks much more manageable written as a series. The theorem can be summarized as
Example 2 APPLYNG THE BINOMIAL THEOREM Write the binomial expansion of (x + y) 9. Solution Now evaluate each of the binomial coefficients.
Example 2 APPLYNG THE BINOMIAL THEOREM
kth Term of the Binomial Expansion The kth term of the binomial expansion of (x + y) n, where n ≥ k – 1, is
kth Term of the Binomial Expansion To find the kth term of the binomial expansion, use the following steps. Step 1 Find k – 1.This is the exponent on the second part of the binomial. Step 2 Subtract the exponent found in Step 1 from n to get the exponent on the first part of the binomial. Step 3 Determine the coefficient by using the exponents found in the first two steps and n.
Example 5 FINDING A PARTICULAR TERM OF A BINOMIAL EXPANSION Find the seventh term of (a + 2b) 10. Solution In the seventh term, 2b has an exponent of 6 while a has an exponent of 10 – 6 or 4. The seventh term is