Pascal’s Triangle and the Binomial Theorem, then Exam!

1. 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.

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

3. Pascal’s Triangle If you arrange the values of 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.

4. The Binomial Theorem For any positive integer n, the expansion of is: + Note that each term has the form where r is an integer from 0 to n. Examples:

5. Example 1 Expand ( ( )4 )4. a a b b + + SOLUTION In , the power is n 4. So, the coefficients of the terms are the numbers in the 4th row of Pascal’s Triangle. = 1a4b0 4a3b1 6a2b2 4a1b3 1a0b4 ( )4 + + + = a b + + a4 4a3b 6a2b2 4ab3 b4 = + + + + Expand a Power of a Simple Binomial Sum Coefficients: 1, 4, 6, 4, 1 Powers of a: a4, a3, a2, a1, a0 Powers of b: b0, b1, b2, b3, b4

6. Example 2 Expand ( )3. x 5 + SOLUTION Use the binomial theorem with a x and b5. = = ( 3C0x350 3C1x251 3C2x152 3C3x053 3 x + 5 ( = + + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + + 1 x3 1 3 x2 5 3 x1 1 x0 125 25 = x3 15x2 75x 125 = + + + Expand a Power of a Binomial Sum

7. Example 3 ( ( Expand . ) 4 )4 2x 2x – – y y SOLUTION First rewrite the difference as a sum: [ ] ( 4 2x – y ( = + Then use the binomial theorem with a2x and b–y. = = = = [ ] ( ( ( ( ( 4C0 4C1 4 4 0 3 1 2x – y – y – y ( ( ( 2x 2x + ( ( = + + ( ( ( ( 4C2 4C3 2 2 1 3 – y – y ( ( 2x 2x ( + ( + ( ( 4C4 0 4 – y ( 2x ( Expand a Power of a Binomial Difference

8. Example 3 + + + = ( ) ( ) 4 2x – y3 + 16x4 32x3y 24x2y2 8xy3 y4 – = + + + ( ) ( ) ( ) ( ) 16x4 8x3 y 4x2 – ( ( ( ( ( ( ) ) ) ) ) ) 4 6 1 1 1 1 ( ( ) ) y2 y4 Expand a Power of a Binomial Difference

9. Checkpoint Expand the power of the binomial sum or difference. ( 1. )5 a + b ANSWER a5 5a4b 10a3b2 10a2b3 5ab4 b5 + + + + + ( 2. )4 x + 2 ANSWER x4 8x3 24x2 32x 16 + + + + ( 3. )3 3x + 5 27x3 135x2 225x 125 ANSWER + + + Expand a Power of a Binomial Sum or Difference

10. Checkpoint Expand the power of the binomial sum or difference. 4. ( )3 – p 4 p3 12p2 48p 64 ANSWER – – + ( 5. )4 – m n m4 4m3n 6m2n2 4mn3 n4 ANSWER – – + + ( 6. )3 3s – t ANSWER 27s3 27s2t 9st2 t3 – – + Expand a Power of a Binomial Sum or Difference

11. Conclusions Summary Assignment 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. Exit Slip: • How can you calculate the coefficients of the terms of ? • Each term in the expansion of has the form , where r is an integer from 0 to n.

12. Exam on the Fundamental Counting Principle 45 MinutesNo talking – Read Rubric – Read Directions – Good Luck!