Mastering Binomial Theorem: Pascal's Triangle and Coefficients

1. College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson

2. Sequences and Series 9

3. The Binomial Theorem 9.6

4. Binomial • An expression of the form a + b is called a binomial. • Although in principle it’s easy to raise a + b to any power, raising it to a very high power would be tedious. • Here, we find a formula that gives the expansion of (a + b)n for any natural number n and then prove it using mathematical induction.

5. Expanding (a + b)n

6. Expanding (a + b)n • To find a pattern in the expansion of (a + b)n, we first look at some special cases:

7. Expanding (a + b)n • The following simple patterns emerge for the expansion of (a + b)n: • There are n + 1 terms, the first being anand the last bn. • The exponents of a decrease by 1 from term to term while the exponents of b increase by 1. • The sum of the exponents of a and b in each term is n.

8. Expanding (a + b)n • For instance, notice how the exponents of a and b behave in the expansion of (a + b)5. • The exponents of a decrease. • The exponents of b increase.

9. Expanding (a + b)n • With these observations, we can write the form of the expansion of (a + b)n for any natural number n. • For example, writing a question mark for the missing coefficients, we have: • To complete the expansion, we need to determine these coefficients.

10. Expanding (a + b)n • To find a pattern, let’s write the coefficients in the expansion of (a + b)n for the first few values of n in a triangular array, which is called Pascal’s triangle.

11. Pascal’s Triangle • The row corresponding to (a + b)0is called the zeroth row. • It is included to show the symmetry of the array.

12. Key Property of Pascal’s Triangle • The key observation about Pascal’s triangle is the following property. • Every entry (other than a 1) is the sum of the two entries diagonally above it. • From this property, it’s easy to find any row of Pascal’s triangle from the row above it.

13. Key Property of Pascal’s Triangle • For instance, we find the sixth and seventh rows, starting with the fifth row:

14. Key Property of Pascal’s Triangle • To see why this property holds, let’s consider the following expansions:

15. Key Property of Pascal’s Triangle • We arrive at the expansion of (a + b)6by multiplying (a + b)5 by (a + b). • Notice, for instance, that the circled term in the expansion of (a + b)6 is obtained via this multiplication from the two circled terms above it.

16. Key Property of Pascal’s Triangle • We get this when the two terms above it are multiplied by b and a, respectively. • Thus, its coefficient is the sum of the coefficients of these two terms.

17. Key Property of Pascal’s Triangle • We will use this observation at the end of the section when we prove the Binomial Theorem. • Having found these patterns, we can now easily obtain the expansion of any binomial, at least to relatively small powers.

18. E.g. 1—Expanding a Binomial Using Pascal’s Triangle • Find the expansion of (a + b)7 using Pascal’s triangle. • The first term in the expansion is a7, and the last term is b7. • Using the fact that the exponent of a decreases by 1 from term to term and that of b increases by 1 from term to term, we have:

19. E.g. 1—Expanding a Binomial Using Pascal’s Triangle • The appropriate coefficients appear in the seventh row of Pascal’s triangle. • Thus,

20. E.g. 2—Expanding a Binomial Using Pascal’s Triangle • Use Pascal’s triangle to expand (2 – 3x)5 • We find the expansion of (a + b)5 and then substitute 2 for a and –3x for b. • Using Pascal’s triangle for the coefficients, we get:

21. E.g. 2—Expanding a Binomial Using Pascal’s Triangle • Substituting a = 2 and b = –3x gives:

22. The Binomial Coefficients

23. The Binomial Coefficients • Although Pascal’s triangle is useful in finding the binomial expansion for reasonably small values of n, it isn’t practical for finding (a + b)n for large values of n. • The reason is that the method we use for finding the successive rows of Pascal’s triangle is recursive. • Thus, to find the 100th row of this triangle, we must first find the preceding 99 rows.

24. The Binomial Coefficients • We need to examine the pattern in the coefficients more carefully to develop a formula that allows us to calculate directly any coefficient in the binomial expansion. • Such a formula exists, and the rest of the section is devoted to finding and proving it. • However, to state this formula, we need some notation.

25. n factorial • The product of the first n natural numbers is denoted by n!and is called n factorial:n! = 1 · 2 · 3 · … · (n – 1) · n

26. 0 factorial • We also define 0! as follows: 0! = 1 • This definition of 0! makes many formulas involving factorials shorter and easier to write.

27. The Binomial Coefficient • Let n and r be nonnegative integers with r ≤n. • The binomial coefficientis denoted by and is defined by:

28. Example (a) E.g. 3—Binomial Coefficients

29. Example (b) E.g. 3—Binomial Coefficients

30. Example (c) E.g. 3—Binomial Coefficients

31. Binomial Coefficients • Although the binomial coefficient is defined in terms of a fraction, all the results of Example 3 are natural numbers. • In fact, is always a natural number. • See Exercise 54.

32. Binomial Coefficients • Notice that the binomial coefficients in parts (b) and (c) of Example 3 are equal. • This is a special case of the following relation. • You are asked to prove this in Exercise 52.

33. Binomial Coefficients • To see the connection between the binomial coefficients and the binomial expansion of (a + b)n, let’s calculate these binomial coefficients:

34. Binomial Coefficients • These are precisely the entries in the fifth row of Pascal’s triangle. • In fact, we can write Pascal’s triangle as follows.

35. Binomial Coefficients

36. Binomial Coefficients • To demonstrate that this pattern holds, we need to show that any entry in this version of Pascal’s triangle is the sum of the two entries diagonally above it. • That is, we must show that each entry satisfies the key property of Pascal’s triangle. • We now state this property in terms of the binomial coefficients.

37. Key Property of the Binomial Coefficients • For any nonnegative integers r and k with r ≤k, • The two terms on the left side are adjacent entries in the kth row of Pascal’s triangle. • The term on the right side is the entry diagonally below them, in the (k + 1)st row.

38. Key Property of the Binomial Coefficients • Thus, this equation is a restatement of the key property of Pascal’s triangle in terms of the binomial coefficients. • A proof of this formula is outlined in Exercise 53.

39. The Binomial Theorem

40. The Binomial Theorem • We prove this at the end of the section. • First, let’s look at some of its applications.

41. E.g. 4—Expanding a Binomial Using Binomial Theorem • Use the Binomial Theorem to expand (x + y)4 • By the Binomial Theorem,

42. E.g. 4—Expanding a Binomial Using Binomial Theorem • Verify that: • It follows that:

43. E.g. 5—Expanding a Binomial Using Binomial Theorem • Use the Binomial Theorem to expand • We first find the expansion of (a + b)8. • Then, we substitute for a and –1 for b.

44. E.g. 5—Expanding a Binomial Using Binomial Theorem • Using the Binomial Theorem, we have:

45. E.g. 5—Expanding a Binomial Using Binomial Theorem • Verify that: • Thus,

46. E.g. 5—Expanding a Binomial Using Binomial Theorem • Performing the substitutions a =x1/2 and b = –1 gives:

47. E.g. 5—Expanding a Binomial Using Binomial Theorem • This simplifies to:

48. General Term of the Binomial Expansion • The Binomial Theorem can be used to find a particular term of a binomial expansion without having to find the entire expansion. • The term that contains arin the expansion of (a + b)n is:

49. E.g. 6—Finding a Particular Term in a Bin. Expansion • Find the term that contains x5in the expansion of (2x + y)20. • The term that contains x5 is given by the formula for the general term with: a = 2x, b =y, n = 20, r = 5

50. E.g. 6—Finding a Particular Term in a Bin. Expansion • So, this term is: