Section 8.5 The Binomial Theorem

The Binomial Theorem 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

The Binomial Theorem Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 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

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 = 1x2 + 2xy + 1y2 (x + y)3 = 1x3 + 3x2y + 3xy2 + 1y3 See the Triangle?

The Binomial Theorem When expanding a binomial remember: (x + y ) 4 The first term of the binomial will decrease in powers. The second term of the binomial will increase in powers. (x + y ) 4 _x4 +_ x3y + _x2y2 + _xy3 +_ y4 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. First Term Second Term = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4 The operation between the two terms goes with the second term. Now simplify where possible. = x4 + 4x3y + 6x2y2 + 4xy3 + y4

The Binomial Theorem Expand (x – 3y)3. x is the first term and negative 3y is the second term (x – 3y)3 _x3 + _x2(–3y) + _x(–3y)2 + _(–3y)3 = 1x3 + 3x2(–3y) + 3x(–3y)2 + 1(–3y)3 = 1x3 +3x2(–3y) + 3x(9y2) + 1(– 27y3) = x3 – 9x2y + 27xy2 – 27y3

The Binomial Theorem The second method is expanding by the Binomial Theorem. Before beginning one needs to know the notation nCr. Example: Find the value of 5C3 Can you find how to do 5C3 on the calculator?

The Binomial Theorem The Binomial Theorem In the expansion of (x + y)n (x + y)n = xn +nxn–1y + … +nCrxn–ryr + … + nxyn–1 + yn The coefficient of xn–ryr is The symbol is often used in place of nCr to denote binomial coefficients.

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.

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

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.

The Binomial Theorem What you should know: 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.