Term 1 : Unit 3

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

2. Binomial Theorem 3.1 The Binomial Expansion of (1 + b) n Objectives 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,

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

4. Binomial Theorem A cuboid with volume ab2 Another cuboid with volume a2b A cube of side a + b Split the cube up as shown A small cube with volume a3 A cuboid with volume a2b 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.

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. Write ascending powers of b from b0 to b6.

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

8. Binomial Theorem Take care of the minus signs here. Notice how the signs alternate between odd and even terms.

9. Binomial Theorem Remember to include the coefficients inside the parentheses.

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

11. Binomial Theorem The binomial coefficientcan be found from this formula. The number of terms in the numerator and denominator is always the same. r! – r factorial

12. Binomial Theorem The Binomial Theorem

13. Binomial Theorem Example 3 . Using this result

14. Binomial Theorem Example 5 . Using this result

15. Binomial Theorem Exercise 6.1, qn 3(d), (g)

16. Binomial Theorem 3.2 The Binomial Expansion of (a + b) n Objectives 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 .

17. Binomial Theorem The Binomial Theorem

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. 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. 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 Be careful with negative values.

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

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