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This printable supports Common Core Mathematics Standard HSA-APR.C.5

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## Binomial Theorem

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

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