Expansion of Binomials

1
Expansion of Binomials
2
(xy)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 (xy)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 (xy)n without the laborious task of
  successive multiplications of (xy)

7
(xy)1

• (xy)1xy
• 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
(xy)1

• (xy)1xy
• 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?

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(xy)2

• (xy)(xy)

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(xy)2(xy)(xy)
Write the first expression twice for the two
terms in the second expression
x y x y


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(xy)2(xy)(xy)
Place each term of the second expression below
x y x y
x x y y

12
(xy)2(xy)(xy)
Multiply down the columns, then combine like terms
x y x y
x x y y
x2 xy xy y2
13
(xy)2
x22xyy2
14
(xy)2

• (xy)2x22xyy2
• 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
(xy)2

• (xy)2x22xyy2
• 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
(xy)2

• (xy)2x22xyy2
• 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
(xy)3

• (xy)3(xy)2(xy)

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(xy)3(x22xyy2)(xy)
Write the first expression twice for the two
terms in the second expression
x2 2xy y2 x2 2xy y2


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(xy)3(x22xyy2)(xy)
Place each term of the second expression below
x2 2xy y2 x2 2xy y2
x x x y y y

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(xy)3(x22xyy2)(xy)
Multiply down the columns, then combine like terms
x2 2xy y2 x2 2xy y2
x x x y y y
x3 2x2y xy2 x2y 2xy2 y3
21
(xy)3

• (xy)3x33x2y3xy2y3

22
(xy)3

• (xy)3x33x2y3xy2y3

• 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
(xy)3

• (xy)3x33x2y3xy2y3

• 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
(xy)3

• (xy)3x33x2y3xy2y3

• 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
(xy)4

• (xy)4(xy)3(xy)

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(xy)4(x33x2y3xy2y3)(xy)
Write the first expression twice for the two
terms in the second expression
x3 3x2y 3xy2 y3 x3 3x2y 3xy2 y3


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(xy)4(x33x2y3xy2y3)(xy)
Place each term of the second expression below
x3 3x2y 3xy2 y3 x3 3x2y 3xy2 y3
x x x x y y y y

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(xy)4(x33x2y3xy2y3)(xy)
Multiply down the columns, then combine like terms
x3 3x2y 3xy2 y3 x3 3x2y 3xy2 y3
x x x x y y y y
x4 3x3y 3x2y2 xy3 x3y 3x2y2 3xy3 y4
29
(xy)4

• (xy)4x44x3y6x2y24xy3y4

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(xy)4

• (xy)4x44x3y6x2y24xy3y4

• 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?

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(xy)4

• (xy)4x44x3y6x2y24xy3y4

• 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
(xy)4

• (xy)4x44x3y6x2y24xy3y4

• 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 n1
• sum of exponents in each term n
• exponent of x decreases from n to 0
• exponent of y increases from 0 to n

n degree terms sum of exponents
1 1 2 1
2 2 3 2
3 3 4 3
4 4 5 4
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Pattern of coefficients degrees 1 to 4
degree coefficients
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
What is the pattern from row to row?
35
Coefficients of 5th degree expansion
degree coefficients
4 1 4 6 4 1
5 1 5 10 10 5 1



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This pattern of coefficients is called Pascals
Triangle

• It can be extended to find the coefficients of
  any degree expansion of a binomial

37
(xy)5

• What is the degree of the expansion?
• How many terms are in the expansion?

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(xy)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?

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(xy)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?

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(xy)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

• (xy)5
• x55x4y10x3y210x2y35xy4y5

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Would you want to build Pascals Triangle for
(xy)99?

• You could, but it would be a large triangle.
• Is there a short cut?
• Yes, indeed there is!

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Factorials
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n!

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

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• 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
(xy)n

• Binomial Theorem
• For r 0 to n
• The (r1)th term is
• n!/(n-r)!r!x(n-r)yr

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(xy)7

• n7
• r0
• 011st term
• 7!/(7-0)!0!x(7-0)y0
• 7!/7!x7y0
• x7

48
(xy)7

• n7
• r1
• 112nd term
• 7!/(7-1)!1!x(7-1)y1
• 7!/6!x6y1
• 76!/6!x6y1
• 7x6y

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(xy)7

• n7
• r2
• 213rd term
• 7!/(7-2)!2!x(7-2)y2
• 7!/5!2!x5y2
• 765!/5!2!x5y2
• 76/2x5y2
• 73x5y2
• 21x5y2

50
(xy)7

• n7
• r3
• 314th term
• 7!/(7-3)!3!x(7-3)y3
• 7!/4!3!x4y3
• 7654!/4!3!x4y3
• 765/321x4y3
• 75x4y3
• 35x4y3

51
(xy)7

• n7
• r4
• 415th term
• 7!/(7-4)!4!x(7-4)y4
• 7!/3!4!x3y4
• 7654!/3!4!x3y4
• 765/321x3y4
• 75x3y4
• 35x3y4

52
(xy)7

• n7
• r5
• 516th term
• 7!/(7-5)!5!x(7-5)y5
• 7!/2!5!x2y5
• 765!/2!5!x2y5
• 76/21x2y5
• 73x2y5
• 21x2y5

53
(xy)7

• n7
• r6
• 617th term
• 7!/(7-6)!6!x(7-6)y6
• 7!/1!6!x1y6
• 76!/6!x1y6
• 7xy6

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(xy)7

• n7
• r7
• 718th term
• 7!/(7-7)!7!x(7-7)y7
• 7!/0!7!x0y7
• 7!/7!x0y7
• y7

55
(xy)7

• x77x6y21x5y235x4y3
• 35x3y421x2y57xy6y7

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What about x a number
57
(x6)4

• x44x3(6)16x2(6)24x(6)3(6)4
• x424x3216x2864x1296

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Knowing the binomial theorem

• Can help you factor polynomials

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8x336x254x27

• (xy)3x33x2y3xy2y3
• Is the first term a cube?
• 8x3(2x)3
• Is the fourth term a cube?
• 27(3)3
• Divide the second term by the binomial
  coefficient 3
• Is the coefficient the cube root of the first
  term squared times the cube root of the fourth
  term?
• 36x2?312x2(4x2)(3)(2x)2(3)
• Divide the third term by the binomial coefficient
  3
• Is the coefficient the cube root of the first
  term times the cube root of the fourth term
  squared?
• 54x?318x(2x)(9)(2x)(3)2
• The polynomial is the cube of a binomial
• 8x336x254x27(2x3)3

60
Conclusions

• The degree of the expansion is the exponent the
  binomial is raised to
• The number of terms in the expansion is one more
  than the degree of the expansion
• Pascals Triangle gives the binomial coefficients
  of the expansion
• The binomial theorem is useful for large
  expansions

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Questions