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Copyright © Cengage Learning. All rights reserved. 9 Sequences, Series, and Probability.

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1 Copyright © Cengage Learning. All rights reserved. 9 Sequences, Series, and Probability

2 9.5 Copyright © Cengage Learning. All rights reserved. THE BINOMIAL THEOREM

3 3 Use the Binomial Theorem to calculate binomial coefficients. Use Pascals Triangle to calculate binomial coefficients. Use binomial coefficients to write binomial expansions. What You Should Learn

4 4 Binomial Coefficients

5 5 Recall that a binomial is a polynomial that has two terms. In this section, you will study a formula that provides a quick method of raising a binomial to a power. To begin, look at the expansion of (x + y) n for several values of n. (x + y) 0 = 1 (x + y) 1 = x + y (x + y) 2 = x 2 + 2xy + y 2 (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 (x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 (x + y) 5 = x 5 + 5x 4 y + 10x 3 y x 2 y 3 + 5xy 4 + y 5

6 6 Binomial Coefficients There are several observations you can make about these expansions. 1. In each expansion, there are n + 1 terms. 2. In each expansion, x and y have symmetrical roles. The powers of x decrease by 1 in successive terms, whereas the powers of y increase by The sum of the powers of each term is n. For instance, in the expansion of (x + y) 5, the sum of the powers of each term is 5. (x + y) 5 = x 5 + 5x 4 y x 3 y x 2 y 3 + 5x 1 y 4 + y = = 5

7 7 Binomial Coefficients 4. The coefficients increase and then decrease in a symmetric pattern. The coefficients of a binomial expansion are called binomial coefficients. To find them, you can use the Binomial Theorem.

8 8 Example 1 – Finding Binomial Coefficients Find each binomial coefficient. a. 8 C 2 b.c. 7 C 0 d. Solution: a.

9 9 Example 1 – Solution b. contd

10 10 Example 1 – Solution c. d. contd

11 11 Binomial Coefficients When r 0 and r n, as in parts (a) and (b) above, there is a simple pattern for evaluating binomial coefficients that works because there will always be factorial terms that divide out from the expression.

12 12 Binomial Coefficients In general, it is true that n C r = n C n – r. This shows the symmetric property of binomial coefficients.

13 13 Pascals Triangle

14 14 Pascals Triangle There is a convenient way to remember the pattern for binomial coefficients. By arranging the coefficients in a triangular pattern, you obtain the following array, which is called Pascals Triangle.

15 15 Pascals Triangle The first and last numbers in each row of Pascals Triangle are 1. Every other number in each row is formed by adding the two numbers immediately above the number.

16 16 Pascals Triangle Pascal noticed that numbers in this triangle are precisely the same numbers that are the coefficients of binomial expansions, as follows.

17 17 Pascals Triangle The top row in Pascals Triangle is called the zeroth row because it corresponds to the binomial expansion (x + y) 0 = 1. Similarly, the next row is called the first row because it corresponds to the binomial expansion (x + y) 1 = 1(x) + 1(y). In general, the nth row in Pascals Triangle gives the coefficients of (x + y) n.

18 18 Example 3 – Using Pascals Triangle Use the seventh row of Pascals Triangle to find the binomial coefficients. 8 C 0, 8 C 1, 8 C 2, 8 C 3, 8 C 4, 8 C 5, 8 C 6, 8 C 7, 8 C 8 Solution:

19 19 Binomial Expansions

20 20 Binomial Expansions When you write out the coefficients for a binomial that is raised to a power, you are expanding a binomial. The formulas for binomial coefficients give you an easy way to expand binomials, as demonstrated in the next example.

21 21 Example 4 – Expanding a Binomial Write the expansion of the expression (x + 1) 3. Solution: The binomial coefficients from the third row of Pascals Triangle are 1, 3, 3, 1. So, the expansion is as follows. (x + 1) 3 = (1)x 3 + (3)x 2 (1) + (3)x(1 2 ) + (1)(1 3 ) = x 3 + 3x 2 + 3x + 1

22 22 Binomial Expansions To expand binomials representing differences rather than sums, you alternate signs. Here are two examples. (x – 1) 3 = x 3 – 3x 2 + 3x – 1 (x – 1) 4 = x 4 – 4x 3 + 6x 2 – 4x + 1 Sometimes you will need to find a specific term in a binomial expansion. Instead of writing out the entire expansion, you can use the fact that, from the Binomial Theorem, the (r + 1)th term is n C r x n – r y r.


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