17Calculus Precalculus - Binomial Theorem
17Calculus
The Binomial Theorem is a technique to expand binomial terms efficiently without having to multiply out and collect terms. There is some notation that you need to learn but it is not a hard technique and will save you time in your calculus work.
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Binomial Theorem
\[ (a+b)^n = \sum_{k=0}^{n}{ \frac{n!}{(n-k)!k!}a^{n-k}b^k } \]
Understanding The Notation
1. A binomial term looks like \(a+b\), i.e. there are exactly 2 terms, a and b, added together and raised to some power.
2. The exponent n must be an integer greater than 1.
3. If you need a refresher on factorials, you can review them on the factorials page.
4. You may also see alternate notation for the factorial term like these.
\[
\frac{n!}{(n-k)!k!}
=
\left(
\begin{array}{c}
n \\ k
\end{array}
\right)
=
{}_{n}C_{k}
\]
The notation \( {}_{n}C_{k} \) is read 'n choose k'.
5. If you are not familiar with summation (sigma) notation, here is a good video that explains how it works.
Khan Academy - Sequences and Series (part 1) [9min-48secs]
In this video he introduces sequences, series and sigma notation. He does a pretty good job explaining the basics required for understanding series.
About 3 minutes into the video, he says something that is correct in this context but not in general. He says that you can add numbers in any order and get the same answer. This is true for a finite list of numbers, like he is doing here, but it is NOT always true if you have an infinite list of numbers. You will probably run across this case in your textbook while you are studying this topic.
video by Khan Academy |
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Understanding and Using The Theorem
Here is a great video explaining, step-by-step, how to use the binomial theorem with an example. When applying the binomial theorem, we sometimes say that we are finding the binomial expansion.
NancyPi - How to Use the Binomial Theorem [19min-58secs]
video by NancyPi |
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Pascal's Triangle
One way to calculate the factorial term is to use Pascal's Triangle. Here is a video explaining this with a couple of examples.
PatrickJMT - Pascal's Triangle and the Binomial Coefficients [5min-40secs]
video by PatrickJMT |
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Okay, time for the practice problems.
Practice
Unless otherwise instructed, find the binomial expansion using the Binomial Theorem.
\((x+2)^5\)
Problem Statement
Find the binomial expansion of \((x+2)^5\) using the Binomial Theorem.
Solution
MIP4U - 2488 video solution
video by MIP4U |
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\((x+y)^5\)
Problem Statement
Find the binomial expansion of \((x+y)^5\) using the Binomial Theorem.
Solution
Thinkwell - 2491 video solution
video by Thinkwell |
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Find the binomial expansion of \((a+b)^5\) and use the result to expand \((x+1)^5\).
Problem Statement
Find the binomial expansion of \((a+b)^5\) and use the result to expand \((x+1)^5\).
Solution
PatrickJMT - 2485 video solution
video by PatrickJMT |
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\((a+5b)^3\)
Problem Statement
Find the binomial expansion of \((a+5b)^3\) using the Binomial Theorem.
Solution
Thinkwell - 2492 video solution
video by Thinkwell |
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\((x-4)^5\)
Problem Statement
Find the binomial expansion of \((x-4)^5\) using the Binomial Theorem.
Solution
MIP4U - 2487 video solution
video by MIP4U |
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\((2x-3)^4\)
Problem Statement
Find the binomial expansion of \((2x-3)^4\) using the Binomial Theorem.
Solution
MIP4U - 2489 video solution
video by MIP4U |
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\((3x-y)^3\)
Problem Statement
Find the binomial expansion of \((3x-y)^3\) using the Binomial Theorem.
Solution
PatrickJMT - 2486 video solution
video by PatrickJMT |
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\((2t-s)^5\)
Problem Statement
Find the binomial expansion of \((2t-s)^5\) using the Binomial Theorem.
Solution
Brian McLogan - 2496 video solution
video by Brian McLogan |
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\((3x^2-5y)^4\)
Problem Statement
Find the binomial expansion of \((3x^2-5y)^4\) using the Binomial Theorem.
Solution
MIP4U - 2490 video solution
video by MIP4U |
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Find the third term of the binomial expansion of \((2x+3)^4\).
Problem Statement
Find the third term of the binomial expansion of \((2x+3)^4\).
Solution
Thinkwell - 2493 video solution
video by Thinkwell |
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Determine the third term of the expansion of \((3x+1)^8\).
Problem Statement
Determine the third term of the expansion of \((3x+1)^8\).
Solution
Brian McLogan - 2495 video solution
video by Brian McLogan |
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Find the fifth term of the expansion of \((2x-3y)^7\).
Problem Statement
Find the fifth term of the expansion of \((2x-3y)^7\).
Solution
Brian McLogan - 2494 video solution
video by Brian McLogan |
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