## 17Calculus Precalculus - Binomial Theorem

### Functions

Functions

Polynomials

Rational Functions

Matrices

Systems

Trigonometry

Complex Numbers

### Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

### Articles

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.

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.

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

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

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

### 2488 video

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

$$(x+y)^5$$

Problem Statement

Find the binomial expansion of $$(x+y)^5$$ using the Binomial Theorem.

Solution

### 2491 video

video by Thinkwell

Log in to rate this practice problem and to see it's current rating.

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

### 2485 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

$$(a+5b)^3$$

Problem Statement

Find the binomial expansion of $$(a+5b)^3$$ using the Binomial Theorem.

Solution

### 2492 video

video by Thinkwell

Log in to rate this practice problem and to see it's current rating.

$$(x-4)^5$$

Problem Statement

Find the binomial expansion of $$(x-4)^5$$ using the Binomial Theorem.

Solution

### 2487 video

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

$$(2x-3)^4$$

Problem Statement

Find the binomial expansion of $$(2x-3)^4$$ using the Binomial Theorem.

Solution

### 2489 video

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

$$(3x-y)^3$$

Problem Statement

Find the binomial expansion of $$(3x-y)^3$$ using the Binomial Theorem.

Solution

### 2486 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

$$(2t-s)^5$$

Problem Statement

Find the binomial expansion of $$(2t-s)^5$$ using the Binomial Theorem.

Solution

### 2496 video

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

$$(3x^2-5y)^4$$

Problem Statement

Find the binomial expansion of $$(3x^2-5y)^4$$ using the Binomial Theorem.

Solution

### 2490 video

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

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

### 2493 video

video by Thinkwell

Log in to rate this practice problem and to see it's current rating.

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

### 2495 video

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

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

### 2494 video

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

Really UNDERSTAND Precalculus

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

### Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

 The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.