1. Binomial Theorem Binomial Theorem Term 1 : Unit 3
3.1 The Binomial Expansion of (1 + b) n
Binomial Theorem
6.1 The Binomial Expansion of (1 + b) n
6.2 The Binomial Expansion of (a + b) n
3.2 The Binomial Expansion of (a + b) n

2. Binomial Theorem 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,
6.1 The Binomial Expansion of (1 + b) n
Objectives
In this lesson you will use Pascal’s triangle or n C r 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. You will identify and find a particular term in the expansion (1 + b) n using the result Tr+1 n C r b r .

3. Binomial Theorem Separate the component parts.
Split the square in four.
A square of side
( a + b ).
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Binomial expansion for n = 2.

4. Binomial Theorem A cuboid with volume ab2
Page 119
A cuboid with volume ab2
Another cuboid with volume a2b
Split the cube up as shown
A cube of side a + b
A cuboid with volume a2b
A small cube with volume a3
And another
Finally, a cube of volume b3.
And another
And another

5. Binomial Theorem Pascal’s Triangle
Add two adjacent terms to make the term below.
Now, we will apply the triangle to the binomial expansion.
Page 120

6. Binomial Theorem Using Pascal’s Triangle to expand (1 + b) 6
Take the 6th row of Pascal’s Triangle.
Use these numbers as coefficients.
Form into a series.
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Write ascending powers of b from b0 to b6.

7. Binomial Theorem Example 1 .
Write ascending powers of b from b0 to b5.
Example 1, Page 121
Use these numbers as coefficients.
Take the 5th row of Pascal’s Triangle.

8. Binomial Theorem Take care of the minus signs here.
Example 1, Page 121
Notice how the signs alternate between odd and even terms.

9. Remember to include the coefficients inside the parentheses.
Binomial Theorem
Remember to include the coefficients inside the parentheses.
Example 1, Page 121

10. n is the row and r is the position (counting from 0).
Binomial Theorem
The fifth row of Pascal’s Triangle was
Using Binomial Coefficient notation, these numbers are
n is the row and r is the position (counting from 0).
Example Page 121

11. Binomial Theorem
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
Example Page 121

12. Binomial Theorem
The Binomial Theorem
Example Page 122

13. Binomial Theorem
Example 3 .
Using this result
Example 3, Page 123

14. Binomial Theorem
Example 5 .
Using this result
Example 5, Page 124

15. Binomial Theorem Exercise 6.1, qn 3(d), (g)
Exercise 6.1, Page 124, Question 3 (d) and (g)

16. Binomial Theorem Objectives 3.2 The Binomial Expansion of (a + b) n
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
6.1 The Binomial Expansion of (1 + b) n
Objectives
In this lesson you will use Pascal’s triangle or n C r 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. You will identify and find a particular term in the expansion (1 + b) n using the result Tr+1 n C r b r .

17. Binomial Theorem
The Binomial Theorem
Example Page 125

18. Binomial Theorem Example The combined total of powers is always 5.
Write descending powers of a from a5 to a0.
Write ascending powers of b from b0 to b5.
Example
Take the 6th row of Pascal’s Triangle.
Use these numbers as coefficients.

19. Binomial Theorem Example 6(b)
Don’t try to simplify yet – not until the next stage.
Example 6, Page 126
Notice that the third term is independent of x.

20. Binomial Theorem Example 8(b) There is no need to find all the terms.
Looking at the combined powers of x
Example 8, Page 127
Be careful with negative values.

21. There is no need to find all the terms.
Binomial Theorem
Exercise 6.2, qn 6(b)
There is no need to find all the terms.
Question 6 (b) Exercise 6.2, Page 128

22. The combined powers of x are 0.
Binomial Theorem
Exercise 6.2, qn 7(c)
The combined powers of x are 0.
Question 7 (c) Exercise 6.2, Page 128