6.8

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6.8 Pascals Triangle and the Binomial Theorem
2
The Binomial Theorem
Strategy only how do we expand these? 1. (x
2)2 2. (2x 3)2 3. (x 3)3 4. (a b)4
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The Binomial Theorem
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THAT is a LOT of work!

• Isnt there an easier way?

5
Introducing Pascals Triangle

• Take a moment to copy the first 6 rows. What
  patterns do you see?

Row 5
Row 6
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The Binomial Theorem
Use Pascals Triangle to expand (a b)5.
Use the row that has 5 as its second number.
In its simplest form, the expansion is a5 5a4b
10a3b2 10a2b3 5ab4 b5.
Row 5
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The Binomial Theorem
Use Pascals Triangle to expand (x 3)4.
First write the pattern for raising a binomial to
the fourth power.
Since (x 3)4 (x (3))4, substitute x for a
and 3 for b.
(x (3))4 x4 4x3(3) 6x2(3)2 4x(3)3
(3)4
x4 12x3 54x2 108x 81
The expansion of (x 3)4 is x4 12x3 54x2
108x 81.
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The Binomial Theorem
Use the Binomial Theorem to expand (x y)9.
Write the pattern for raising a binomial to the
ninth power. (a b)9 9C0a9 9C1a8b 9C2a7b2
9C3a6b3 9C4a5b4 9C5a4b5 9C6a3b6
9C7a2b7 9C8ab8 9C9b9
Substitute x for a and y for b. Evaluate each
combination. (x y)9 9C0x9 9C1x8(y)
9C2x7(y)2 9C3x6(y)3 9C4x5(y)4
9C5x4(y)5 9C6x3(y)6 9C7x2(y)7
9C8x(y)8 9C9(y)9
x9 9x8y 36x7y2 84x6y3 126x5y4
126x4y5 84x3y6 36x2y7 9xy8 y9
The expansion of (x y)9 is x9 9x8y 36x7y2
84x6y3 126x5y4 126x4y5 84x3y6 36x2y7
9xy8 y9.
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Lets Try Some

• Expand the following
• a) (x-y5)3 b) (3x-2y)4

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Lets Try Some

• Expand the following
• (x-y5)3

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Lets Try Some

• Expand the following
• (3x-2y)4

12
Lets Try Some

• Expand the following
• (3x-2y)4

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How does this relate to probability?

• You can use the Binomial Theorem to solve
  probability problems. If an event has a
  probability of success p and a probability of
  failure q, each term in the expansion of (p q)n
  represents a probability.

Example 10C2 p8 q2 represents the probability
of 8 successes in 10 tries
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The Binomial Theorem
Brianna makes about 90 of the shots on goal she
attempts. Find the probability that Bri makes
exactly 7 out of 12 consecutive goals.
Since you want 7 successes (and 5 failures), use
the term p7q5. This term has the coefficient
12C5.
Probability (7 out of 10) 12C5 p7q5
0.0037881114 Simplify.
Bri has about a 0.4 chance of making exactly 7
out of 12 consecutive goals.