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Section 15.5: The Binomial Theorem Pascal’s Triange

Section 15.5: The Binomial Theorem Pascal’s Triange. Pre-Calculus. Learning Targets. Define and apply the Binomial Theorem Define and apply Pascal’s Triangle Communicate the relationship between the Binomial Theorem and Pascal’s Triangle. Problem 1.

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Section 15.5: The Binomial Theorem Pascal’s Triange

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  1. Section 15.5: The Binomial TheoremPascal’s Triange Pre-Calculus

  2. Learning Targets • Define and apply the Binomial Theorem • Define and apply Pascal’s Triangle • Communicate the relationship between the Binomial Theorem and Pascal’s Triangle

  3. Problem 1 • In your groups, take 5 minutes to expand the following: • 1. • 2. • 3.

  4. Problem 1

  5. GOal • Our goal today is to figure out a system that will allows us to quickly expand anything of the form • Let’s start by seeing if we can recognize a pattern from the binomials we expanded earlier.

  6. Problem 1 Let’s write out the coefficients of each term (a+b)0 1 (a+b)1 1 1 (a+b)2 1 2 1 (a+b)3 1 3 3 1

  7. Problem 1 (a+b)01 (a+b)1 1 1 (a+b)2 1 2 1 (a+b)3 1 3 3 1 What do you notice about each row to the next?

  8. Pascal’s Triangle • This relationship is called Pascal’s Triangle. • Pascal’s triangle is a simple array but has many unusual relationships among the numbers. • It’s quite fascinating!

  9. Pascal’s Triangles Relationships • 1. To construct the triangle, you add two adjacent numbers together to get the number below it • 2. Diagonals: • 1st diagonal = 1’s • 2nd diagonal = counting numbers • 3rd diagonal = triangular numbers • Etc.

  10. Pascal’s Triangles Relationships • 3. Odds & Evens: If you color the odd and even numbers you come up with the Sierpinski Triangle which is a fractal! • 4. Horizontal Sums:

  11. Pascal’s Triangles Relationships • 5. Exponents of 11 • 6. Squares

  12. Pascal’s Triangles Relationships • 7. Fibonacci Sequence

  13. Pascal’s Triangles Relationships • 8. Binomial Coefficients • Each row of Pascal’s Triangle relates to the coefficients of an expanded binomial. (a+b)01 (a+b)1 1 1 (a+b)2 1 2 1 (a+b)3 1 3 3 1 (a+b)4 1 4 6 4 1

  14. Pascal’s Triangle Relationships • 9. Combinations • Let’s look at the 3rd row: = 1 3 3 1 Notice there is only one way to get . Thus, 3C3 where I am choosing all 3 a’s. To get this means that there are 2 a’s and 1 b. Thus there are 3C2 ways to get this term in relation to the a To get , 3C1 in relation to the a To get , 3C0 in relation to the a Thus, the number also represent different combinations too! WHOA!

  15. Binomial Theorem • Putting relationship 8 and 9 together. We get the formula for the binomial theorem • Where is another way to represent combinations (n choose k)

  16. Example 1 • Let’s put this all together. • Expand (a + b)4 • By the binomial theorem • Thus, • Then, the combinations are determined by Pascal’s Triangle or you can actually solve them

  17. Example 2 • Write the first 4 terms in the expansion of

  18. Example 3 • What is the coefficient of in the expansion of ?

  19. Homework • Textbook Pg 592 (written exercises) #3, 14, 19, 22

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