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Binomial Theorem

Binomial Theorem. Objectives: To use the Binomial Theorem and Pascal’s Triangle to expand powers of binomials. Assignment Binomial Theorem Supplement. Powers of Binomials. Let’s say that you wanted to find (1 + x ) 5 .

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Binomial Theorem

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  1. Binomial Theorem Objectives: • To use the Binomial Theorem and Pascal’s Triangle to expand powers of binomials Assignment • Binomial Theorem Supplement

  2. Powers of Binomials Let’s say that you wanted to find (1 + x)5. You could expand it out and then use the distributive property until your head hurt. Or you could take a shortcut, which involves something called the Binomial Theorem. = Headache

  3. Symmetrical Patterns • Notice that the expansion of a binomial is symmetrical. • There is one more term than the power.

  4. Symmetrical Patterns • The sum of the powers of each term is the power of the binomial. 3+1=4 1+3=4 2+2=4

  5. Symmetrical Patterns • As the power of x decreases by one, the power of y increases by one. • The question is, how do we get those symmetrical, binomial coefficients? Blaise has our answer.

  6. Blaise Pascal • 1623-1662 • Also known as “Snake Eyes” (Not really) • Father of Probability Theory • Came up with an easy way to find binomial coefficients (Pascal: Avid stamp collector)

  7. Pascal’s Triangle Each row will begin and end with the number 1.

  8. Pascal’s Triangle Each row will begin and end with the number 1.

  9. Pascal’s Triangle Each row will begin and end with the number 1.

  10. Pascal’s Triangle Each row will begin and end with the number 1.

  11. Pascal’s Triangle Each row will begin and end with the number 1.

  12. Pascal’s Triangle To get the numbers between the ones, add the two numbers directly above them.

  13. Pascal’s Triangle To get the numbers between the ones, add the two numbers directly above them.

  14. Pascal’s Triangle To get the numbers between the ones, add the two numbers directly above them.

  15. Pascal’s Triangle To get the numbers between the ones, add the two numbers directly above them.

  16. Pascal’s Triangle To get the numbers between the ones, add the two numbers directly above them.

  17. Pascal’s Triangle To get the numbers between the ones, add the two numbers directly above them.

  18. Pascal’s Triangle Can you finish the triangle without clicking ahead?

  19. Pascal’s Triangle Can you finish the triangle without clicking ahead?

  20. Pascal’s Triangle Can you finish the triangle without clicking ahead?

  21. Pascal’s Triangle Can you finish the triangle without clicking ahead?

  22. Pascal’s Triangle The triangle, of course, would not stop at the 7th power. It would continue forever, or at least until your pencil broke, your hand went dead, or you went crazy.

  23. Pascal’s Triangle This is how you use it to expand binomials:

  24. Example 1 Expand (x + 5)4.

  25. What About Differnces? When you expand a difference instead of a sum, the signs alternate, starting with +.

  26. Example 2 Expand (x – 2)5.

  27. Example 3 Expand (2x – 3)3.

  28. Binomial Theorem Objectives: • To use the Binomial Theorem and Pascal’s Triangle to take powers of complex numbers Assignment • Homework Supplement

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