College Algebra

1

• College Algebra
• Sixth Edition
• James Stewart ? Lothar Redlin ? Saleem Watson

2

• Sequences and Series

8
3
8.6

• The Binomial Theorem

4
Binomial

• An expression of the form a b is called a
  binomial.
• Although in principle its 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.

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• Expanding (a b)n

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Expanding (a b)n

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

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Expanding (a b)n

• The following simple patterns emerge for the
  expansion of (a b)n
• There are n 1 terms, the first being an and
  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.

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Expanding (a b)n

• To find a pattern, lets write the coefficients
  in the expansion of (a b)n for the first few
  values of n in a triangular array, which is
  called Pascals triangle.

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Pascals Triangle

• The row corresponding to (a b)0 is called the
  zeroth row.
• It is included to show the symmetry of the array.

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Key Property of Pascals Triangle

• The key observation about Pascals triangle is
  the following property.
• Every entry (other than a 1) is the sum of the
  two entries diagonally above it.
• From this property, its easy to find any row of
  Pascals triangle from the row above it.

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Key Property of Pascals Triangle

• For instance, we find the sixth and seventh rows,
  starting with the fifth row

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Key Property of Pascals Triangle

• To see why this property holds, lets consider
  the following expansions

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Key Property of Pascals Triangle

• We arrive at the expansion of (a b)6 by
  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.

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Key Property of Pascals 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.

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Key Property of Pascals 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.

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E.g. 1Expanding a Binomial Using Pascals
Triangle

• Find the expansion of (a b)7 using Pascals
  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

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E.g. 1Expanding a Binomial Using Pascals
Triangle

• The appropriate coefficients appear in the
  seventh row of Pascals triangle.
• Thus,

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E.g. 2Expanding a Binomial Using Pascals
Triangle

• Use Pascals 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 Pascals triangle for the coefficients, we
  get

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E.g. 2Expanding a Binomial Using Pascals
Triangle

• Substituting a 2 and b 3x gives

22

• The Binomial Coefficients

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The Binomial Coefficients

• Although Pascals triangle is useful in finding
  the binomial expansion for reasonably small
  values of n, it isnt 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 Pascals triangle is
  recursive.
• Thus, to find the 100th row of this triangle, we
  must first find the preceding 99 rows.

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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.

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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

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0 factorial

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

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The Binomial Coefficient

• Let n and r be nonnegative integers with r n.
• The binomial coefficient is denoted by and
  is defined by

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E.g. 3Calculating Binomial Coefficients
Example (a)

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Example (b)
E.g. 3Calculating Binomial Coefficients

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Example (c)
E.g. 3Calculating Binomial Coefficients

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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.

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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.

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Binomial Coefficients

• To see the connection between the binomial
  coefficients and the binomial expansion of (a
  b)n, lets calculate these binomial coefficients

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Binomial Coefficients

• These are precisely the entries in the fifth row
  of Pascals triangle.
• In fact, we can write Pascals triangle as
  follows.

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Binomial Coefficients
36
Binomial Coefficients

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

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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 Pascals triangle.
• The term on the right side is the entry
  diagonally below them, in the (k 1)st row.

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Key Property of the Binomial Coefficients

• Thus, this equation is a restatement of the key
  property of Pascals triangle in terms of the
  binomial coefficients.
• A proof of this formula is outlined in Exercise
  53.

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• The Binomial Theorem

40
The Binomial Theorem

• We prove this at the end of the section.
• First, lets look at some of its applications.

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E.g. 4Expanding a Binomial Using Binomial Theorem

• Use the Binomial Theorem to expand (x y)4
• By the Binomial Theorem,

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E.g. 4Expanding a Binomial Using Binomial Theorem

• Verify that
• It follows that

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E.g. 5Expanding a Binomial Using the 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.

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E.g. 5Expanding a Binomial Using the Binomial
Theorem

• Using the Binomial Theorem, we have

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E.g. 5Expanding a Binomial Using the Binomial
Theorem

• Verify that
• So,

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E.g. 5Expanding a Binomial Using the Binomial
Theorem

• Performing the substitutions a x1/2 and b
  1 gives

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E.g. 5Expanding a Binomial Using the 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 ar in the expansion of (a
  b)n is

49
E.g. 6Finding a Particular Term in a Binomial
Expansion

• Find the term that contains x5 in 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. 6Finding a Particular Term in a Binomial
Expansion

• So, this term is

51
E.g. 7Finding a Particular Term in a Binomial
Expansion

• Find the coefficient of x8 in the expansion of
• Both x2 and 1/x are powers of x.
• So, the power of x in each term of the expansion
  is determined by both terms of the binomial.

52
E.g. 7Finding a Particular Term in a Binomial
Expansion

• To find the required coefficient, we first find
  the general term in the expansion.
• By the formula, we have a x2, b 1/x, n
  10
• So, the general term is

53
E.g. 7Finding a Particular Term in a Binomial
Expansion

• Thus, the term that contains x8 is the term in
  which 3r 10 8
• r 6
• So, the required coefficient is

54

• Proof of the Binomial Theorem

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Binomial Theorem

• We now give a proof of the Binomial Theorem
  using mathematical induction.

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Binomial TheoremProof

• Let P(n) denote the statement

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Binomial TheoremProof

• Step 1 We show that P(1) is true.
• However, P(1) is just the statement which is
  certainly true.

58
Binomial TheoremProof

• Step 2 We assume that P(k) is true.
• Thus, our induction hypothesis is
• We use this to show that P(k 1) is true.

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Binomial TheoremProof
60
Binomial TheoremProof

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Binomial TheoremProof

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Binomial TheoremProof

63
Binomial TheoremProof

• Using the key property of the binomial
  coefficients, we can write each of the
  expressions in square brackets as a single
  binomial coefficient.

64
Binomial TheoremProof

• Also, writing the first and last coefficients as
  (these are equal to 1 by Exercise
  50) gives the following result.

65
Binomial TheoremProof

• However, this last equation is precisely P(k
  1).
• This completes the induction step.

66
Binomial TheoremProof

• Having proved Steps 1 and 2, we conclude, by the
  Principle of Mathematical Induction, that the
  theorem is true for all natural numbers n.