### About The Binomial Theorem:

The binomial theorem gives us a way to quickly expand certain binomials raised to a natural number.

Test Objectives

- Demonstrate the ability to use factorial notation
- Demonstrate the ability to evaluate a combination
- Demonstrate the ability to find the expansion of a binomial
- Demonstrate the ability to find the kth term of a binomial expansion

#1:

Instructions: Evaluate each expression.

$$a)\hspace{.2em}{10 \choose 2}$$

$$b)\hspace{.2em}{9 \choose 7}$$

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#2:

Instructions: Expand completely.

$$a)\hspace{.2em}(x - 2y)^4$$

$$b)\hspace{.2em}(2x + y)^6$$

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#3:

Instructions: Expand completely.

$$a)\hspace{.2em}(y - 3x)^5$$

$$b)\hspace{.2em}(2x - y)^6$$

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#4:

Instructions: Expand completely.

$$a)\hspace{.2em}(3y + x)^5$$

Instructions: Find each term described.

$$b)\hspace{.2em}(x - 2y)^7$$ 4^{th} term in expansion

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#5:

Instructions: Find each term described.

$$a)\hspace{.2em}(y - 2x)^6$$ 2^{nd} term in expansion

$$b)\hspace{.2em}(x + 2y)^7$$ 6^{th} term in expansion

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Written Solutions:

#1:

Solutions:

a) 45

b) 36

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#2:

Solutions:

a) x^{4} - 8x^{3}y + 24x^{2}y^{2} - 32xy^{3} + 16y^{4}

b) 64x^{6} + 192x^{5}y + 240x^{4}y^{2} + 160x^{3}y^{3} + 60x^{2}y^{4} + 12xy^{5} + y^{6}

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#3:

Solutions:

a) y^{5} - 15y^{4}x + 90y^{3}x^{2} - 270y^{2}x^{3} + 405yx^{4} - 243x^{5}

b) 64x^{6} - 192x^{5}y + 240x^{4}y^{2} - 160x^{3}y^{3} + 60x^{2}y^{4} - 12xy^{5} + y^{6}

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#4:

Solutions:

a) 243y^{5} + 405y^{4}x + 270y^{3}x^{2} + 90y^{2}x^{3} + 15yx^{4} + x^{5}

b) -280x^{4}y^{3}

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#5:

Solutions:

a) -12y^{5}x

b) 672x^{2}y^{5}