2 Pascal’s TriangleWith 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 mathematicianBlaise Pascal. It was, however, known long before his time.
3 Pascal’s TriangleTo 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 isUsing these coefficients, we obtain the expansion of (x + y)6 :
4 n-FactorialAlthough 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.
6 Binomial Coefficients Now look at the coefficients of the expansionThe coefficient of the second term, 5x4y, is 5, and the exponents on the variablesare 4 and 1. Note that
7 Binomial CoefficientFor nonnegative integers n and r, withr ≤ n,
8 Binomial Coefficients The binomial coefficients are numbers fromPascal’s triangle. For example, is the firstnumber 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.
9 d. Solution EVALUATING BINOMIAL COEFFICIENTS Example 1 Evaluate the binomial coefficient.d.Solution
10 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 xn, and the last term is yn.3. In each succeeding term, the exponent on x decreases by 1 and the exponent on y increases by 1.4. The sum of the exponents on x and y in any term is n.5. The coefficient of the term with xryn – r orxn – r y r is
11 Binomial TheoremFor any positive integer n and any complex numbers x and y,
12 Note The binomial theorem looks much more manageable written as a series. The theorem can be summarized as
13 Solution APPLYNG THE BINOMIAL THEOREM Example 2 Write the binomial expansion of (x + y)9.SolutionNow evaluate each of the binomial coefficients.
15 kth Term of the Binomial Expansion The kth term of the binomial expansion of (x + y)n, where n ≥ k – 1, is
16 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.
17 Solution FINDING A PARTICULAR TERM OF A BINOMIAL EXPANSION Example 5 Find the seventh term of (a + 2b)10.SolutionIn the seventh term, 2b has an exponent of 6 while a has an exponent of 10 – 6 or 4. The seventh term is