Binomial Expansions using Pascals Triangle

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Binomial Expansionsusing Pascals Triangle

• How to find any term in the expansion (ab)n

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Pascals Triangle
11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6
15 20 15 6 11 7 21 35 35 21 7 11 8 28 56 70 56
28 8 11 9 36 84 126 126 84 36 9 1
Click me!
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Algebraic expansion
Expand (1x)2
(1 x)2 (1 x)(1 x) 1 2x x2
Expand (1x)3
(1 x)3 (1 x)(1 x)(1 x) (1 x)(1 2x
x2) 1 3x 3x2 x3
What do you notice??
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Algebraic expansion
The coefficients are lines from Pascals
Triangle!!
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Using Pascals Triangle
11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 1
(ab)0 (ab)1 (ab)2 (ab)3 (ab)4
(ab)5
1 1a 1b 1a2 2ab 1b2 1a3 3a2b 3ab2
1b3 1a4 4a3b 6a2b2 4ab3 1b4 1a5 5a4b
10a3b2 10a2b3 5ab4 1b5
As the power of a goes down by 1, the power of b
goes up by 1
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Example 1
Expand (13x)4
a 1 b 3x
(ab)4 1a4 4a3b 6a2b2 4ab3 1b4
1a4 4a3b 6a2b2 4ab3 1b4
1 4(1)3(3x) 12x 6(1)2(3x)2 6(9x2)
54x2 4(1)(3x)3 4(27x3) 108x3 1(3x)4 81x4
(13x)4 1 12x 54x2 108x3 81x4
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Example 2 - have a try
Expand (1-2x)3
a 1 b -2x
(ab)3 1a3 3a2b 3ab2 1b3
1a3 3a2b 3ab2 1b3
1 3(1)2(-2x) -6x 3(1) (-2x)2 3(4x2)
12x2 1(-2x)3 -8x3
(1-2x)3 1 - 6x 12x2 - 8x3
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Activity

• Turn to page 201 in your book.
• Answer questions 2 and 3 in Exercise 6F