©2001 by R. Villar All Rights Reserved 9.5 The Binomial Theorem ©2001 by R. Villar All Rights Reserved

The Binomial Theorem Recall that a binomial has two terms... (x + y) The Binomial Theorem gives us a quick method to expand binomials raised to powers such as… (x + y)0 (x + y)1 (x + y)2 (x + y)3 Study the following… 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Row 0 This triangle is called Pascal’s Triangle (named after mathematician Blaise Pascal). Row 1 Row 2 Notice that row 5 comes from adding up row 4’s adjacent numbers. (The first row is named row 0). Row 3 Row 4 Row 5 What will row 6 be? Row 6 1 6 15 20 15 6 1 This pattern will help us find the coefficients when we expand binomials...

(x + y)0 = (x + y)1 = (x + y)2 = (x + y)3 = (x + y)4 = (x + y)5 = Study the following... (x + y)0 = (x + y)1 = (x + y)2 = (x + y)3 = (x + y)4 = (x + y)5 = 1 x + y x2 + 2xy + y2 x3 + 3x2y + 3xy2 + y3 x4 + 4x3y + 6x2y2 + 4xy3 + y4 x5 + 5x4y +10x3y2 +10x2y3+5xy4 + y5 (x + y)6 = x6+6x5y+15x4y2+20x3y3+15x2y4+6xy5 + y6 Notice that for (x + y)n there are n + 1 terms… Notice that the binomial coefficients follow Pascal’s Triangle… Notice that the first term is xn and the last term is yn … Notice that the sum of the exponents of each term is n… Use this pattern to expand (x + y)6...

The binomial expansion of (x + y)n is The Binomial Theorem: The binomial expansion of (x + y)n is (x + y)n = xn + nxn–1 y + … + n! xn-m ym +…+ nxyn–1 + yn (n – m)!m! The coefficient xn – my m is denoted by This can be used to find any coefficient for the expansion of any binomial... Example. Find the binomial coefficients for : Row 6 of Pascal’s Triangle looks like this: 1 6 15 20 15 6 1 Here is another way to find the value of this coefficient = 6! 4!2! = 6 • 5 • 4! 4! • 2 • 1 = 15

Example: Find the binomial coefficients for : = 12! 4!8! = 12 • 11 • 10 • 9 • 8! 4 • 3 • 2 • 1 • 8! = 11880 24 = 495

Row 6 of Pascal’s Triangle looks like this: 1 6 15 20 15 6 1 Example: Use Pascal’s Triangle to find the binomial coefficients for the expansion of (x + y)8 . Row 6 of Pascal’s Triangle looks like this: 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Complete the pattern for rows 7 and 8...

Example: Use the binomial Theorem to expand (x – 3)4 Row 4 of Pascal’s Triangle looks like this: 1 4 6 4 1 which translates to: x4 + 4x3y + 6x2y2 + 4xy3 + y4 Replace the y’s with –3... x4 + 4x3(–3) + 6x2(–3)2 + 4x(–3)3 + (–3)4 Simplify... x4 – 12x3+ 54x2 – 108x + 81

Example: Use the binomial Theorem to expand (2x + 5)5 Row 5 of Pascal’s Triangle looks like this: 1 5 10 10 5 1 which translates to: x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 Replace the x’s with 2x and the y’s with +5... (2x)5 +5(2x)4(5)+10(2x)3(5)2 +10(2x)2(5)3 +5(2x)(5)4 +(5)5 Simplify… 32x5 +5(16x4)(5)+10(8x3)(25) +10(4x2)(125) +5(2x)(625) + 3125 32x5 + 400x4 + 2000x3 + 5000x2 + 6250x + 3125