6.8 The Binomial Theorem 2011
February 04, 2011
6.8 The Binomial Theorem
Objectives:
• Use Pascal's Triangle.
• Use the Binomial Theorem.
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6.8 The Binomial Theorem 2011
Evaluate.
3. 5C0
Warm­up
2. (2x ­ 3)2
4. 5C1
n
Cr = 5. 5C2
n!
r!(n ­ r)!
http://www.adonald.btinternet.co.uk/Factor/Zero.html
Multiply.
1. (x + 2)2
February 04, 2011
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6.8 The Binomial Theorem 2011
February 04, 2011
Binomial Expansion
Expand the following binomials in your notes.
1. (a + b)2
2. (a + b)3
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6.8 The Binomial Theorem 2011
February 04, 2011
Set down your pen/pencil and look at Pascal's Triangle.
What do you notice?
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6.8 The Binomial Theorem 2011
February 04, 2011
Pascal's Triangle
Pascal's Triangle: A triangular array of numbers formed by lining the border with 1's, and then placing the sum of the two adjacent numbers within a row between and underneath the two original numbers.
*Named after a French mathematician named Blaise Pascal (1623 ­ 1662).
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6.8 The Binomial Theorem 2011
February 04, 2011
Pascal's Triangle
Write it down in your notes now (if you haven't already)!
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6.8 The Binomial Theorem 2011
February 04, 2011
Example #1: Use Pascal's Triangle to expand (a + b)6.
(Hint: Use the row that has 6 as its second number.)
The exponents for a begin with 6 and decrease to 0.
1a6b0 + 6a5b1 + 15a4b2 + 20a3b3 + 15a2b4 + 6a1b5 + 1a0b6 The exponents for b begin with 0 and increase to 6.
*Now write the answer in simplified form.
a6 + 6a5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab5 + b6 7
6.8 The Binomial Theorem 2011
February 04, 2011
Example #2: Use Pascal's Triangle to expand (a + b)7.
(Hint: Use the row that has 7 as its second number.)
The exponents for a begin with 7 and decrease to 0.
0 7
6
1a7b0 + 7a6b1 + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7a1b + 1a b
The exponents for b begin with 0 and increase to 7.
Simplified Form:
a7 + 7a6b + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7ab6 + b7
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6.8 The Binomial Theorem 2011
February 04, 2011
Example #3: Use Pascal's Triangle to expand (x ­ 2)3.
Step 1: Write the pattern for (a + b)3.
1a3b0 + 3a2b1 + 3a1b2 + 1a0b3
Step 2: Plug in x for a and ­2 for b.
1x3(­2)0 + 3x2(­2)1 + 3x1(­2)2 + 1x0(­2)3
Step 3: Simplify.
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6.8 The Binomial Theorem 2011
February 04, 2011
The Binomial Theorem
Using combinations (instead of Pascal's triangle) to help find the terms of a binomial expansion.
Example: (a + b)4
4C0, 4C1, 4C2, 4C3, 4C4 OR
1 4 6 4 1
0 4
b a
1
+ b
1a b + 4a b + 6a b + 4a
4 0
3 1
2 2
1 3
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6.8 The Binomial Theorem 2011
February 04, 2011
Example #4: Use the Binomial Theorem to expand (g + h)5.
Step 1: Write the pattern for (g + h)9.
5
C0g5h0 + 5C1g4h1 + 5C2g4h2 + 5C3g2h3 + 5C4g1h4 + 5C5g0h5
Step 2: Simplify each combination nCr.
0 5
1g5h0 + 5g4h1 + 10g4h2 + 10g2h3 + 5g1h4 + 1g h
Step 3: Simplify.
g5 + 5g4h + 10g4h2 + 10g2h3 + 5gh4 + h5
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6.8 The Binomial Theorem 2011
February 04, 2011
Why is the Binomial Theorem HANDY?
Example #5: Find the third term of (x + 4)12.
n
Ct x
n ­ t
(4)
t
n = 12
t = 3 ­ 1 (for third term)
2
10
(4)
x
C
12 2 66 x10(4)
264 x10
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6.8 The Binomial Theorem 2011
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6.8 The Binomial Theorem 2011
February 04, 2011
Why is the Binomial Theorem HANDY?
Example #6: Assume that Chauncey's probability for success on any free throw is the same as his cumulative record to date (89.3%). Find the probability that he will make exactly 6 out of 10 Chauncey Billups
consecutive free throws. Step 1: Find the correct term and coefficient. p6q4
6 successes & 4 failures
Coefficient 10C4
Step 2: Calculate.
P(6 out of 10) = 10C4 p6q4
= 10C4 (0.893)6(0.107)4
= 210 (0.893)6(0.107)4
= 0.01395928 ≈ 1.4% chance
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6.8 The Binomial Theorem 2011
February 04, 2011
Homework:
page 355 (2 ­ 22 even, 48 ­ 56 even)
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