1. Section 8.5 The Binomial Theorem

2. In this section you will learn two techniques for expanding a binomial when raised to a power. The first method is called expanding by using Pascal’s Triangle. Read about Pascal’s Triangle and Expanding a Binomial at http://www.themathpage.com/aPrecalc/binomial-theorem.htm#pascal

3. The Binomial Theorem Pascal’s Triangle 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Can you figure out the pattern? What would be the next row? 1 6 15 20 15 6 1

4. The Binomial Theorem Each row of Pascal’s Triangle is the coefficients for the Terms when the binomial is expanded. (x + y) 0 = 1 (x + y) 1 = 1x + 1y (x + y) 2 = 1x 2 + 2xy + 1y 2 (x + y) 3 = 1x 3 + 3x 2 y + 3xy 2 + 1y 3 See the Triangle?

5. The Binomial Theorem When expanding a binomial remember: The first term of the binomial will decrease in powers. The second term of the binomial will increase in powers. (x + y ) 4 First Term Second Term The operation between the two terms goes with the second term. _x 4 +_ x 3 y + _x 2 y 2 + _xy 3 +_ y 4 In Pascal’s Triangle find the row whose second number is the power to be expanded. Here it is 4. Now place the numbers in the row in front of the terms. = 1x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + 1y 4 Now simplify where possible. = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4

6. The Binomial Theorem Expand (x – 3y) 3. x is the first term and negative 3y is the second term (x – 3y) 3 _x 3 + _x 2 (–3y) + _x(–3y) 2 + _(–3y) 3 = 1x 3 + 3x 2 (–3y) + 3x(–3y) 2 + 1(–3y) 3 = 1x 3 +3x 2 (–3y) + 3x(9y 2 ) + 1(– 27y 3 ) = x 3 – 9x 2 y + 27xy 2 – 27y 3

7. The Binomial Theorem The second method is expanding by the Binomial Theorem. Before beginning one needs to know the notation n C r. Example: Find the value of 5 C 3 Can you find how to do 5 C 3 on the calculator?

8. The Binomial Theorem In the expansion of (x + y) n (x + y) n = x n +nx n–1 y + … + n C r x n–r y r + … + nxy n–1 + y n The coefficient of x n–r y r is The symbol is often used in place of n C r to denote binomial coefficients.

9. The Binomial Theorem Expand (x + y) 3 using the Binomial Theorem. Notice in line 1 that n stays constant and r decreases by 1 each time. Also, notice the sum of the exponents is the same as the exponent of the binomial being expanded.

10. The Binomial Expression Expand (2x – y) 5 using the Binomial Theorem.

11. The Binomial Theorem Sometimes only a certain term of the binomial expansion needs to be found. In that case remember n stays constant and the term to be found uses the formula r + 1. Find the fourth term of (x +2y) 6. Solution: Here n = 6 and since r + 1 = 4, then r = 3.

12. The Binomial Theorem What you should know: 1.How to apply Pascal’s Triangle to expand a binomial. 2. How to apply the Binomial Theorem to expand a binomial. 3. How to find a certain term of an expanded binomial.