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.
$$b)\hspace{.2em}(x - 2y)^7$$ 4th term in expansion
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#5:
Instructions: Find each term described.
$$a)\hspace{.2em}(y - 2x)^6$$ 2nd term in expansion
$$b)\hspace{.2em}(x + 2y)^7$$ 6th term in expansion
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Written Solutions:
#1:
Solutions:
a) 45
b) 36
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#2:
Solutions:
a) x4 - 8x3y + 24x2y2 - 32xy3 + 16y4
b) 64x6 + 192x5y + 240x4y2 + 160x3y3 + 60x2y4 + 12xy5 + y6
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#3:
Solutions:
a) y5 - 15y4x + 90y3x2 - 270y2x3 + 405yx4 - 243x5
b) 64x6 - 192x5y + 240x4y2 - 160x3y3 + 60x2y4 - 12xy5 + y6
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#4:
Solutions:
a) 243y5 + 405y4x + 270y3x2 + 90y2x3 + 15yx4 + x5
b) -280x4y3
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#5:
Solutions:
a) -12y5x
b) 672x2y5
