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5.8 – Binomial Theorem

Key Terms

  • Binomial Coefficient – The coefficient of a given term in a binomial expansion.
    • The coefficient of the k-term in the expansion of (x+a)^n is denoted \binom{n}{k}.
  • Binomial Expansion – An expression that is equivalent to a binomial raised to a positive integer power, and that results from carrying out the multiplication indicated by the power.
  • Combinations – Unique selections of items, no repeats.
    • Ex. If you have 3 different fruits: apple, pear, and banana, and must choose only 2, you wouldn’t choose [apple and pear], then [pear and apple] because those would be repeats.  The combination of apple and pear will only happen once, no matter the order.
      • Choosing 2, you can also have a pear with banana or an apple with banana.
  • Binomial Theorem – A polynomial identity which states: (x+y)^n=\binom{n}{0}x^n+ \binom{n}{1}x^{n-1}y^1+ \binom{n}{2}x^{n-2}y^2+ \binom{n}{3}x^{n-3}y^3+...+ \binom{n}{n}y^n
  • Factorial – The product of n and all positive whole numbers less than n.
    • The notation n! means “n factorial.”
    • Ex. 4! = 4 3 2 1 = 24
    • Ex. 8! = 8  4 3 2 1 = 40,320
    • There is a button on your calculator “n!” you can use!
  • Pascal’s Triangle – A triangular arrangement of numbers in which each row gives you the coefficients of a binomial expansion.
    • Any number in the triangle can be computed by taking the sum of the two adjacent numbers in the previous row.
  • Permutation – A selection of objects in which the order of the objects matters.
    • Use this to show how many ways can you arrange a certain number of items.
    • \frac{n!}{k!(n-k)!}
    • 0! = 1

 


Notes

  • Pascal’s Triangle
    • Do you see a diagonal pattern?
      • 1 + 1 = 2
      • 1 + 2 = 3
      • 1 + 3 = 4, so the coefficient of the 2nd term in the 4th line down will be 4.
      • What will the 2nd term look like including the variables?  Look for the pattern. Try writing it out on paper!  Scroll ALL the way down to see the answer!
    • Do you see a horizontal pattern (for the exponents)?
      • From left to right, powers of x get smaller while powers of y get bigger
      • The row will always start and end with the same power (which matches the row that it is in)
    • The pattern found with Pascal’s triangle works for any binomial
    • Using Pascal’s triangle may be faster than using the FOIL method when binomials have coefficients.

Alg2B 5.8 Higher Power Pyramid

 

  • Watch and copy the video below

 

  • Ex. (2+y)^3
    • Look at the 3rd row of Pascal’s Triangle and replace ALL x’s with 2’s.
    • Substitute 2 for x:  1(2)^3+3(2)^2y+3(2)y^2+1y^3
    • Simplified: 8+12y+6y^2+y^3

 

  • Ex. (3x-2y)^2
    • Rewrite as: (3x+(-2y))^2
    • Substitute 3x for x and substitute -2y for y: 1(3x)^2+2(3x)(-2y)+(-2y)^2
    • Simplify: 9x^2-12xy+4y^2

 

  • Pascal’s Patterns Enumerated
    1. Expanding a binomial to the nth power gives n + 1 terms.
    2. The first term of (x+y)^n will always be x^n, and the last will always be y^n.
    3. The sum of the exponents of each term of the expansion of (x+y)^n is n.
    4. The exponents of the x-terms in the expansion of (x+y)^n will start at n and decrease by 1 for every consecutive term.
    5. The exponents of the y-terms in the expansion of (x+y)^n will start at zero and increase by 1 for every consecutive term.
    6. The coefficients follow the pattern of Pascal’s triangle for the nth row.

 


  • Negative Values for Pascal’s Triangle Expansion
    • For (x-y)^n, raising a negative value to a power creates a pattern of alternating signs.
    • For ((-x)+(-y))^n, if n is an even number, every sign will be positive; but, if n is an odd number, every sign will be negative.

 

  • Ex. (x-y)^n

Alg2B 5.8 Ex x-yn

 


  • Pascal’s Triangle by Row (n) and Column (k)
    • Rows: x
    • Columns: y

 


  • Permutations Using the Combination Formula
    • \binom{n}{k} is read “n choose k,”
    • Formula can be used to find the kth term of the nth row of Pascal’s triangle
    • It is defined by the quotient (ratio) of factorials
      • \binom{n}{k}=\frac{n!}{k!(n-k)!}
      • And for some reason, 0! = 1
      • Use a calculator to find the answers

Alg2B 5.8 Factorials

 

  • Ex. You can choose 3 out of 26 friends to go with you to the movies.  How many possible combinations can you have?
    • Since calculators can’t handle numbers that large without converting to scientific notation, here’s a shortcut!

Alg2B 5.8 kn shortcut

 


Examples

  • For (x+y)^6
    • Ex 1a. Which of the following could NOT be a term in the expansion?
      • 6x^5y
      • 15x^2y^4
      • 20xy
      • 15x^3y^3
      • Answer: since each term’s exponents MUST add up to 6, “20xy” is the one that could NOT be a term in the expansion.
    • Ex 1b. What is the expansion?

Alg2B 5.8 Ex Pascal 6

 

  • Ex 2. Calculate the 5th term for (x-3y)^8
    • The first term will have 8 as the x-exponent and zero as the y-exponent.
    • The fifth term will have 8−4 (so = 4) as the x-exponent and 4 as the y-exponent.
    • The term will be positive because (-3y)^4 is positive.
    • The coefficient from the triangle is 70, so the fifth term is:
    • 70x^4(-3y)^4=(70)(81)x^4y^4=5670x^4y^4

Alg2B 5.8 Ex Pascal 8

 


  • Pascal’s Triangle: The Next Level

Alg2B 5.8 Higher Power Pyramid3

 

  • Coefficients of Pascal’s Triangle

Pascal Triangle

 


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