term 1 : unit 3 binomial theorem 3.1 the binomial expansion of (1 + b) n 3.2 the binomial expansion...
TRANSCRIPT
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Term 1 : Unit 3
Binomial Theorem
3.1 The Binomial Expansion of (1 + b) n
3.2 The Binomial Expansion of (a + b) n
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r
n
Objectives
3.1 The Binomial Expansion of (1 + b) n
In this lesson, you will use Pascal’s triangle or to find the
binomial coefficient of any term. You will use the Binomial Theorem to expand (1 + b)n for positive integer values of n and identify and find a particular term in the expansion (1+ b)n using the
result, .1r
r br
nT
Binomial Theorem
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2b
2a b
A square of side
( a + b ).
2 2 22a b a ab b
Binomial Theorem
a
a b
b
2a
ab
ab
a b
a b
Split the square in four.
Separate the component
parts.
Binomial expansion for n = 2.
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3a b
3b
A cube of side a + b
Binomial Theorem
2ab
2ab
2a b
2ab
2a b
2a b
3a
Split the cube up as
shown
A small cube with volume a3
A cuboid with volume
a2b
Another cuboid with volume a2b
And another
A cuboid with volume
ab2
And another
And another
Finally, a cube of
volume b3.
3
3 2 2 33 3
a b
a a b ab b
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Add two adjacent terms to make the
term below.
Binomial Theorem
111
1 1211 3 3
1 14 4611 55 10 10
11 66 15 152011 35217 72135
1 170 56 28 88 28 5611 126 84 36 99 36 84 126
Pascal’s Triangle
Now, we will apply the
triangle to the binomial
expansion.
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Binomial Theorem
Using Pascal’s Triangle to expand (1 + b) 6
1 14 4611 55 10 10
11 66 15 152011 35217 72135
1 170 56 28 88 28 5611 126 84 36 99 36 84 126
b 2b 3b 4b 6b5b 0b
Write ascending powers of b from b0 to
b6.
Take the 6th row of Pascal’s Triangle.
Use these numbers as coefficients.
Form into a series.
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Binomial Theorem
Take the 5th row of Pascal’s Triangle.
Use these numbers as coefficients.
Write ascending powers of b from b0 to b5.
5Expand 1 b
Example 1
2 3 4 51 b b b b b 5 2 3 4 51 1 5 10 10 5b b b b b b
11 55 10 10
.
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Binomial Theorem
5 2 3 4 5
5
Use the result 1 1 5 10 10 5
to find 1
b b b b b b
b
2 3 4 51 5 10 10 5b b b b b
551 1b b
2 3 4 51 5 10 10 5b b b b b
Take care of the minus signs here.
Notice how the signs alternate
between odd and even terms.
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Binomial Theorem
5 2 3 4 5
5
Use the result 1 1 5 10 10 5
to find 1 2
b b b b b b
x
5 2 3 4 51 2 1 5 2 10 2 10 2 5 2 2x x x x x x
2 3 4 51 10 40 80 80 32x x x x
2 3 4 51 5 2 10 4 10 8 5 16 32x x x x x
Remember to include the coefficients inside the
parentheses.
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Binomial Theorem
11 55 10 10
5
5
5
0
5
4
5
1
5
2
5
3
The fifth row of Pascal’s Triangle was
Using Binomial Coefficient notation, these numbers are
In the expansion of 1 the coefficient of is n r n
b br
n is the row and r is the position
(counting from 0).
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Binomial Theorem
1 2 1
1 2 3 2 1
n n n n n r
r r r r
1 2 1 1, 1,
0 !
n n n n n n n r
n r r
5 5 4 3 10
3 3 2 1
8 8 7 6 5 4 56
5 5 4 3 2 1
The binomial coefficient can be found from this formula.
The number of terms in the numerator and denominator
is always the same.r! – r factorial
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Binomial Theorem
0 1 2 31 0 1 2 3
n nn n n n nb b b b b b
n
2 31 1 21 1
2! 3!
n nn n n n nb nb b b b
The Binomial Theorem
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Binomial Theorem
Using this result
82Find the first four terms in the expansion of 1 x
8 2 32 2 2 28 8 81 1
1 2 3x x x x
2 4 68 7 8 7 61 8
2 1 3 2 1x x x
2 4 61 8 28 56x x x 8
Estimate the value of 1.01
88 21.01 1 0.1
2 4 61 8 0.1 28 0.1 56 0.1 1 0.08 0.002 8 0.000 056 1.082 856
Example 3
.
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Binomial Theorem
Using this result
122 5Find the terms in and in the expansion of 1
2
xx x
212
2 2
x
12
3 3 2 533 993 2 1 3 2
2 2 4
xx x x x
5 3 299 333 2
4 2x x x
212 11
2 1 4
x
233
2x
512
5 2
x
512 11 10 9 8
5 4 3 2 1 2
x 599
4x
12
5 3Find the coefficient of in the expansion of 3 2 12
xx x
5165
4x
Example 5
.
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Binomial Theorem
79 2
Find the first four terms, in ascending powers of in the
expansions of 1 2 and 1 2 .
x
x x
9 2 39 8 9 8 71 2 1 9 2 2 2
2 1 3 2 1x x x x
2 31 18 144 672x x x
2 4 61 14 84 280x x x
7 2 32 2 2 27 6 7 6 51 2 1 7 2 2 2
2 1 3 2 1x x x x
Exercise 6.1, qn 3(d), (g)
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Binomial Theorem
In this lesson, you will use the Binomial Theorem to expand (a + b) n for positive integer values of n. You will identify and find a particular term in
the expansion (a + b) n, using the result .1T .n r rr
na b
r
3.2 The Binomial Expansion of (a + b) n
Objectives
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Binomial Theorem
2 3
1 1 2 3
nn n n n nb b b b
ana a a a
The Binomial Theorem
1 1n n
n nb ba b a a
a a
1 2 2 3 3 1 2 3
n n n n n nn n na b a a b a b a b b
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Binomial Theorem
Take the 6th row of
Pascal’s Triangle.
Use these numbers as coefficients.
Write ascending
powers of b from b0 to
b5.
5Expand .a b
b 5 5 4 3 2 2 3 4 55 10 10 5a b a a b a b a b ab b
11 55 10 10
3b 4b 5b2b3a 2a4a5a a
Write descending powers of a
from a5 to a0.
The combined total of powers is always
5.
Example
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Binomial Theorem
6
2
Find, in descending powers of , the first four terms of
1.
x
xx
6 33
206 15x x
x
6 2 36 5 4 3
2 2 2 2
1 1 1 16 15 20x x x x x
x x x x
Don’t try to simplify yet – not until the
next stage.
Notice that the third term is
independent of x.
Example 6(b)
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Binomial Theorem
Looking at the
combined powers of
x
12
15 2
1 1Find the term in in the expansion of 2 .x
x x
12
2
12 1The general term is 2 .
rr
xr x
15
1For the term in , 12 2 15r r
x 9r
318
1220 8x
x
15
1760
x
9
3
15 2
121 1The term in is 2
9x
x x
There is no need to find all the
terms.
Be careful with negative values.
Example 8(b)
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Binomial Theorem
9Find the fourth term in the expansion of 3 2 .x
684 729 8x
9 6 393 2 3 2
3x x
There is no need to
find all the terms.
6The fourth term is 489888 .x
Exercise 6.2, qn 6(b)
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Binomial Theorem
103
2
2Find the constant term in the expansion of .x
x
1212
64210x
x
The combined
powers of x are 0.
The constant term is 13440.
1032
10 2The general term is
rr
xr x
3 10 2 0r r 6r
6
40 32
10 2The term in is
6x x
x
Exercise 6.2, qn 7(c)