Precalculus

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Precalculus

• 8.5 / 9.5
• The Binomial Theorem

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Recall

• Binomial a polynomial with two terms
• We will be looking at ways to simplify the
  operation (x y)n
• Calculate coefficients
• Write expansions
• Use Pascals Triangle

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Lets start with some expansions
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You can notice some things
Each expansion has n1 terms

• Powers of x decrease by one as powers of y
  increase by one
• For every term in the expansion, the exponents
  add to n
• Coefficients increase and decrease symmetrically

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Binomial TheoremFinds binomial coefficients

• For each term in an expansion, it is easy to
  write x and y with their exponents
• For example, for (x y)4
• x4 nx3y nx2y2 nxy3 y4

x terms go down in degree
y terms go up in degree
What are the ns?
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The Binomial Theorem

• In the expansion (x y)n
• the coefficient of a term in the expansion
• xn ryr is

Look familiar?
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• The coefficient of xn ryr
• Where is r coming from?
• The exponent of x subtracted from n
• This will also be the exponent attached to y

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Another Note

• The symbol is often used in place of

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Example 1Finding Binomial Coefficients

• For (xy)8, find the coefficient attached to the
  term whose x term is degree 6 and y terms is
  degree 2.

• The term would be 28x6y2

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Example 2Find the binomial coefficientThen
write the term

The term is 120x7y3
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Binomial Expansions(x y)n

• First, you must find the coefficients
• Then attach x and y using the appropriate
  exponents, following the pattern mentioned earlier

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Example 3Expanding a Binomial

• Write the expansion of (x 1)3

• Write out the terms

• Find the coefficients

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What about (x y)n?

• The signs inside will alternate ,-,,-
• Example (x 1)3
• x (-1)3
• 1x3 3x2(-11) 3x(-12) 1(-13)
• x3 - 3x2 3x - 1

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Practice

• Write the expansion (2x 3)4

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Finding a specific term from an expansion

• What if you want to know the third term from an
  expansion
• We use the fact that to find a term, (r 1) from
  an expansion, we can use the previously seen
  expression
• We just need to find the value of r

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Example 4

• Find the third term of the expansion of
• (x2 4)3

• What are the values of n and r for this example?

n 3
r 1 3 so r 2

• Evaluate

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Practice

• Find the sixth term of (a 2b)8

1792a3b5
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The Shortcut

• There is another pattern to finding binomial
  coefficients
• Pascals Triangle

Row 0
Row 3
. . .
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How to use the triangle

• What are the coefficients of a binomial expansion
  (x y)8?

Row 8
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How to form the triangle

• The first and last entries of every row are both
  1.
• Every other entry is formed by adding together
  the two numbers above it
• One just off to the left
• One just off to the right

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3 1 2
5 4 1
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Example 5

• Write the expansion of (2x 3)4

• Use the 4th row of Pascals Triangle
• 1, 4, 6, 4, 1

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Homework

• P 624 31 67 every other odd, 84