Pascal’s Triangle and the Binomial Theorem, then Exam! 20.0 Students know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers.

Pascal’s Triangle and the Binomial Theorem Objectives Key Words Relate Pascal’s Triangle to the terms of a Binomial Expansion The Binomial Theorem Pascal’s triangle The arrangement of 𝑛 𝐶 𝑟 in a triangular pattern in which each row corresponds to a value of n. (pg 553, you have to see it to believe it!)

Pascal’s Triangle If you arrange the values of 𝑛 𝐶 𝑟 in a triangular pattern in which each row corresponds to a value of n, you get a pattern called Pascal’s triangle. Turn to page 553.

For any positive integer n, the expansion of 𝑎+𝑏 𝑛 is: The Binomial Theorem For any positive integer n, the expansion of 𝑎+𝑏 𝑛 is: 𝑎+𝑏 𝑛 = 𝑛 𝐶 0 𝑎 𝑛 𝑏 0 + 𝑛 𝐶 1 𝑎 𝑛−1 𝑏 1 + 𝑛 𝐶 2 𝑎 𝑛−2 𝑏 2 +⋯+ 𝑛 𝐶 𝑛 𝑎 0 𝑏 𝑛 Note that each term has the form 𝑛 𝐶 𝑟 𝑎 𝑛−𝑟 𝑏 𝑟 where r is an integer from 0 to n. Examples: 𝑎+𝑏 1 𝑎+𝑏 2 𝑎+𝑏 3

Example 1 Expand ( )4. b a + SOLUTION Expand a Power of a Simple Binomial Sum Expand ( )4. b a + SOLUTION In , the power is n 4. So, the coefficients of the terms are the numbers in the 4th row of Pascal’s Triangle. = ( )4 b a + Coefficients: 1, 4, 6, 4, 1 Powers of a: a 4, a 3, a 2, a 1, a 0 Powers of b: b0, b1, b2, b3, b4 = 1a 4b 0 + ( )4 b a 4a 3b 1 6a 2b 2 1a 0b 4 4a 1b 3 = a 4 + 4a 3b 6a 2b 2 4ab 3 b 4 5

Use the binomial theorem with a x and b 5. = Example 2 Expand a Power of a Binomial Sum Expand ( )3. 5 x + SOLUTION Use the binomial theorem with a x and b 5. = 5 x ( + 3 = 3C0x 350 3C1x 251 3C2x 152 3C3x 053 = + ( ) x 3 1 x 2 3 5 x 1 25 x 0 125 = + x 3 15x 2 75x 125 6

First rewrite the difference as a sum: SOLUTION Example 3 Expand a Power of a Binomial Difference Expand y 2x ( – )4 . First rewrite the difference as a sum: SOLUTION = 2x [ + 4 ( – y ] y – ) 4 = Then use the binomial theorem with a 2x and b –y. 4C3 4C2 4C4 = + 2x [ 4 ( – y ] 4C0 4C1 3 1 2 7

Example 3 = + ( ) 16x 4 1 8x 3 4 y – 4x 2 6 y 2 2x – y 3 y 4 = 16x 4 + Expand a Power of a Binomial Difference = + ( ) 16x 4 1 8x 3 4 y – 4x 2 6 y 2 2x – y 3 y 4 = 16x 4 + – 32x 3y 24x 2y 2 8xy 3 y 4 8

Expand the power of the binomial sum or difference. Checkpoint Expand a Power of a Binomial Sum or Difference Expand the power of the binomial sum or difference. 1. b a ( + )5 ANSWER a 5 + 5a 4b 10a 3b 2 10a 2b 3 5ab 4 b 5 2. 2 x ( + )4 ANSWER x 4 + 8x 3 24x 2 32x 16 3. 5 3x ( + )3 ANSWER 27x 3 + 135x 2 225x 125

Expand the power of the binomial sum or difference. Checkpoint Expand a Power of a Binomial Sum or Difference Expand the power of the binomial sum or difference. 4. 4 p ( )3 – p 3 ANSWER + – 12p 2 48p 64 5. – n m ( )4 m 4 ANSWER + – 4m 3n 6m 2n 2 4mn 3 n 4 6. t 3s ( – )3 27s 3 + – 27s 2t 9st 2 t 3 ANSWER

Conclusions Summary Assignment Exit Slip: How can you calculate the coefficients of the terms of 𝑎+𝑏 𝑛 ? Each term in the expansion of 𝑎+𝑏 𝑛 has the form 𝑛 𝐶 𝑟 𝑎 𝑛−𝑟 𝑏 𝑟 , where r is an integer from 0 to n. Pg 555 #(2,6-13) Write the assignment down, you will work on it after the you finish the exam, early. Get ready for the exam.

45 Minutes No talking – Read Rubric – Read Directions – Good Luck! Exam on the Fundamental Counting Principle 45 Minutes No talking – Read Rubric – Read Directions – Good Luck!