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Expansion of Binomials

Expansion of Binomials. (x+y) n. The expansion of a binomial follows a predictable pattern Learn the pattern and you can expand any binomial. What are we doing?. Expanding binomials of the form (x+y) n Looking for patterns in the expansion of binomials

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Expansion of Binomials

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  1. Expansion of Binomials

  2. (x+y)n • The expansion of a binomial follows a predictable pattern • Learn the pattern and you can expand any binomial

  3. What are we doing? • Expanding binomials of the form (x+y)n • Looking for patterns in the expansion of binomials • Developing a method for expanding binomials

  4. Why are we doing this? • Topic in Intermediate and College Algebra • Necessary in Calculus if not for the Chain Rule

  5. What have we learned before that will help? • Distributive property of real numbers • Multiplying polynomials

  6. How will I know if I have learned this? • You will be able to expand any binomial of the form (x+y)n without the laborious task of successive multiplications of (x+y)

  7. (x+y)1 • (x+y)1=x+y • What is the degree of the expansion? • How many terms are in the expansion? • What is the exponent of x in the first term? • What is the exponent of y in the first term? • What is the sum of the exponents in the first term?

  8. (x+y)1 • (x+y)1=x+y • What is the exponent of x in the second term? • What is the exponent of y in the second term? • What is the sum of the exponents in the second term? • What is the coefficient of the first term? • What is the coefficient of the second term?

  9. (x+y)2 (x+y)(x+y)

  10. (x+y)2=(x+y)(x+y) Write the first expression twice for the two terms in the second expression

  11. (x+y)2=(x+y)(x+y) Place each term of the second expression below

  12. (x+y)2=(x+y)(x+y) Multiply down the columns, then combine like terms

  13. (x+y)2 x2+2xy+y2

  14. (x+y)2 • (x+y)2=x2+2xy+y2 • What is the degree of the expansion? • How many terms are in the expansion? • What is the exponent of x in the first term? • What is the exponent of y in the first term? • What is the sum of the exponents in the first term?

  15. (x+y)2 • (x+y)2=x2+2xy+y2 • What is the exponent of x in the second term? • What is the exponent of y in the second term? • What is the sum of the exponents in the second term? • What is the exponent of x in the third term? • What is the exponent of y in the third term? • What is the sum of the exponents in the third term?

  16. (x+y)2 • (x+y)2=x2+2xy+y2 • How do the exponents of x change from left to right? • How do the exponents of y change from left to right? • What is the coefficient of the first term? • What is the coefficient of the second term? • What is the coefficient of the third term?

  17. (x+y)3 (x+y)3=(x+y)2(x+y)

  18. (x+y)3=(x2+2xy+y2)(x+y) Write the first expression twice for the two terms in the second expression

  19. (x+y)3=(x2+2xy+y2)(x+y) Place each term of the second expression below

  20. (x+y)3=(x2+2xy+y2)(x+y) Multiply down the columns, then combine like terms

  21. (x+y)3 (x+y)3=x3+3x2y+3xy2+y3

  22. (x+y)3 • (x+y)3=x3+3x2y+3xy2+y3 • What is the degree of the expansion? • How many terms are in the expansion? • What is the exponent of x in the first term? • What is the exponent of y in the first term? • What is the sum of the exponents in the first term? • What is the exponent of x in the second term? • What is the exponent of y in the second term? • What is the sum of the exponents in the second term?

  23. (x+y)3 • (x+y)3=x3+3x2y+3xy2+y3 • What is the exponent of x in the third term? • What is the exponent of y in the third term? • What is the sum of the exponents in the third term? • What is the exponent of x in the fourth term? • What is the exponent of y in the fourth term? • What is the sum of the exponents in the fourth term?

  24. (x+y)3 • (x+y)3=x3+3x2y+3xy2+y3 • How do the exponents of x change from left to right? • How do the exponents of y change from left to right? • What is the coefficient of the first term? • What is the coefficient of the second term? • What is the coefficient of the third term?

  25. (x+y)4 (x+y)4=(x+y)3(x+y)

  26. (x+y)4=(x3+3x2y+3xy2+y3)(x+y) Write the first expression twice for the two terms in the second expression

  27. (x+y)4=(x3+3x2y+3xy2+y3)(x+y) Place each term of the second expression below

  28. (x+y)4=(x3+3x2y+3xy2+y3)(x+y) Multiply down the columns, then combine like terms

  29. (x+y)4 (x+y)4=x4+4x3y+6x2y2+4xy3+y4

  30. (x+y)4 • (x+y)4=x4+4x3y+6x2y2+4xy3+y4 • What is the degree of the expansion? • How many terms are in the expansion? • What is the exponent of x in the first term? • What is the exponent of y in the first term? • What is the sum of the exponents in the first term? • What is the exponent of x in the second term? • What is the exponent of y in the second term? • What is the sum of the exponents in the second term?

  31. (x+y)4 • (x+y)4=x4+4x3y+6x2y2+4xy3+y4 • What is the exponent of x in the third term? • What is the exponent of y in the third term? • What is the sum of the exponents in the third term? • What is the exponent of x in the fourth term? • What is the exponent of y in the fourth term? • What is the sum of the exponents in the fourth term? • What is the exponent of x in the fifth term? • What is the exponent of y in the fifth term? • What is the sum of the exponents in the fifth term?

  32. (x+y)4 • (x+y)4=x4+4x3y+6x2y2+4xy3+y4 • How do the exponents of x change from left to right? • How do the exponents of y change from left to right? • What is the coefficient of the first term? • What is the coefficient of the second term? • What is the coefficient of the third term? • What is the coefficient of the fourth term? • What is the coefficient of the fifth term?

  33. Pattern of exponents degrees 1 to 4 • degree of expansion of binomial = n • number of terms in expansion = n+1 • sum of exponents in each term = n • exponent of x decreases from n to 0 • exponent of y increases from 0 to n

  34. Pattern of coefficients degrees 1 to 4 What is the pattern from row to row?

  35. Coefficients of 5th degree expansion

  36. This pattern of coefficients is called Pascal’s Triangle It can be extended to find the coefficients of any degree expansion of a binomial

  37. (x+y)5 • What is the degree of the expansion? • How many terms are in the expansion?

  38. (x+y)5 xy + xy + xy + xy + xy + xy • What is the exponent of x in the first term? • What is the exponent of y in the first term? • What is the exponent of x in the second term? • What is the exponent of y in the second term? • What is the exponent of x in the third term? • What is the exponent of y in the third term?

  39. (x+y)5 xy + xy + xy + xy + xy + xy • What is the exponent of x in the fourth term? • What is the exponent of y in the fourth term? • What is the exponent of x in the fifth term? • What is the exponent of y in the fifth term? • What is the exponent of x in the sixth term? • What is the exponent of y in the sixth term?

  40. (x+y)5 Based on the pattern for binomial coefficients: • What is the binomial coefficient of the first term? • What is the binomial coefficient of the second term? • What is the binomial coefficient of the third term? • What is the binomial coefficient of the fourth term? • What is the binomial coefficient of the fifth term? • What is the binomial coefficient of the sixth term?

  41. (x+y)5= x5+5x4y+10x3y2+10x2y3+5xy4+y5

  42. Would you want to build Pascal’s Triangle for (x+y)99? You could, but it would be a large triangle. Is there a short cut? Yes, indeed there is!

  43. Factorials

  44. n! • Unary operator • Symbol ! • Multiplication of all numbers from n down to 1

  45. 0!=1 • n!=n·(n-1)!=n·(n-1)·(n-2)! • n!/(n-2)!=n·(n-1)·(n-2)!/(n-2)!=n·(n-1) • (nr) means n choose r =n!/(n-r)!r!

  46. (x+y)n • Binomial Theorem For r = 0 to n The (r+1)th term is n!/(n-r)!r!x(n-r)yr

  47. (x+y)7 n=7 r=0 0+1=1st term 7!/(7-0)!0!x(7-0)y0= 7!/7!x7y0= x7

  48. (x+y)7 n=7 r=1 1+1=2nd term 7!/(7-1)!1!x(7-1)y1= 7!/6!x6y1= 7·6!/6!x6y1= 7x6y

  49. (x+y)7 n=7 r=2 2+1=3rd term 7!/(7-2)!2!x(7-2)y2= 7!/5!2!x5y2= 7·6·5!/5!2!x5y2= 7·6/2x5y2= 7·3x5y2= 21x5y2

  50. (x+y)7 n=7 r=3 3+1=4th term 7!/(7-3)!3!x(7-3)y3= 7!/4!3!x4y3= 7·6·5·4!/4!3!x4y3= 7·6·5/3·2·1x4y3= 7·5x4y3= 35x4y3

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