Binomial Theorem (Grades 11-12)

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Binomial Theorem

1. 
How can one determine the coefficients of the Binomial Theorem expansion for [math](x+y)^n[/math], where [math]n[/math] is an integer? There may be more than one correct answer.
  1. Pascal's Triangle
  2. Binomial Permutation
  3. [math]({:(n),(k):}), \ k=0,...,n[/math]
  4. [math](n!)/(k!(n-k)!), \ k=0,...,n[/math]
2. 
Use Pascal's triangle to expand [math](x+y)^3[/math].
  1. [math] (x+y)^3=x^3+y^3 [/math]
  2. [math](x+y)^3=x^1+x^2+x^3+y^1+y^2+y^3 [/math]
  3. [math](x+y)^3=x^3+3x^2y+3xy^2+y^3 [/math]
  4. [math](x+y)^3=xy+3x^2y^2+9x^3y^3 [/math]
3. 
Use Pascal's triangle to expand [math](x+y)^5[/math].
  1. [math] (x+y)^5=x^5+5x^4y+10x^3y^2+10x^2y^3+4xy^4+y^5 [/math]
  2. [math](x+y)^5=x^1+x^2+x^3+x^4+x^5+y^1+y^2+y^3+y^4+y^5 [/math]
  3. [math] (x+y)^5=x^5+y^5 [/math]
  4. [math](x+y)^5=xy+5x^2y^2+10x^3y^3+15x^4y^4+20x^5y^5 [/math]
4. 
Expand the binomial [math](x-3)^4[/math] using Pascal's Triangle.
  1. [math]x^4+12x^3+54x^2+108x+81[/math]
  2. [math]x^4-12x^3+54x^2-108x+81[/math]
  3. [math]x^4-12x^3-54x^2-108x-81[/math]
  4. [math]x^4-12x^3+18x^2-36x+81[/math]
5. 
Find the 7th term in expansion [math](x+y)^8[/math].
  1. [math]8x^2y^6[/math]
  2. [math]28x^2y^6[/math]
  3. [math]70x^2y^6[/math]
  4. [math]56x^2y^6[/math]
6. 
Find the 3rd term in expansion [math](2x-3)^5[/math].
  1. [math]720x^3[/math]
  2. [math]80x^3[/math]
  3. [math]72x^3[/math]
  4. [math]120x^3[/math]
7. 
Approximate [math]1.04^5[/math] with the Binomial Theorem, using only the first 3 terms of the expansion, by rewriting [math]1.04[/math] as [math]1 + 0.04[/math].
  1. 0.016
  2. 1.2
  3. 1.216
  4. 1.21664
8. 
Approximate [math]1.92^7[/math] with the Binomial Theorem, using only the first 3 terms of the expansion, by rewriting [math]1.92[/math] as [math]2 - 0.08[/math].
  1. 92.16
  2. 96.4608
  3. 96.17408
  4. 4.3008
9. 
If using the Binomial Theorem to approximate a rational number of the form [math]a^b[/math], where [math]a>0[/math] is a rational number that is not an integer and [math]b>0[/math] is an integer, will the approximation be less than or greater than the actual value? Explain your answer.



10. 
Euler's number, [math]e[/math], is an irrational number (like [math]pi[/math]). You can approximate [math]e[/math] with the formula [math](1+1/n)^n[/math], such that the approximation improves as [math]n[/math] approaches infinity. Which of the following is an equivalent formula, using the Binomial Theorem and the fact that [math]({:(n),(k):}) (1/n)^k = 1/(k!) [/math] as [math]n[/math] approaches infinity?
  1. [math]sum_{k=0}^n 1/(k!)[/math]
  2. [math]sum_{k=0}^n (k^n)/(k!) [/math]
  3. [math]sum_{k=0}^n (n!)/(k!) [/math]
  4. [math]sum_{k=0}^n (n-k)/(k!)[/math]

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