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23 Nov 2025
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Mastering the Binomial Expansion
Jess
@jess_eqxp
1 / 3
Binomial Expansion Basics
Ever wondered how to expand something like ^8 without multiplying it out eight times? Binomial expansion is your answer, and it works for three main types of expressions.
The binomial theorem gives us a shortcut to expand any binomial raised to a power. Instead of multiplying ... repeatedly, we can use patterns to write out the answer directly.
Pascal's triangle is brilliant for finding coefficients. Each row gives you the numbers you need - just start with 1 at the top, then each number below is the sum of the two numbers above it. Row 4 gives you 1, 4, 6, 4, 1 for expanding something to the power of 4.
When expanding ^n, the powers of 'a' start at n and decrease by 1 each term, whilst the powers of 'b' start at 0 and increase by 1 each term. The coefficients come straight from Pascal's triangle.
Quick tip: The row number in Pascal's triangle matches the power you're expanding to!
Working Through Examples
Let's see binomial expansion in action with some proper examples that show you exactly how it works.
For ^6, you'll use row 6 of Pascal's triangle: 1, 6, 15, 20, 15, 6, 1. Each term follows the pattern: coefficient × x^(decreasing power) × 2^(increasing power). So you get x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64.
The tricky bit comes with negative terms like ^4. Remember that raised to odd powers stays negative, but even powers become positive. This gives you 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4.
Setting up a table can really help you keep track of all the calculations. List your coefficients, then work out each term systematically - don't try to do it all in your head!
Watch out: When you've got constants multiplying your variables (like the 2 in 2x), you need to raise the entire term to the power, not just the variable.
Binomial Coefficients and Combinations
When you need to expand something massive like ^20, Pascal's triangle becomes impractical. That's where binomial coefficients using combinations come to the rescue.
Factorial notation helps us calculate how many ways we can choose objects. The formula for choosing r objects from n objects is nCr = n!/, and this is exactly what gives us our binomial coefficients.
In any binomial expansion ^n, the coefficient of the term a^r × b^ is nCr. This works because you're essentially choosing which r brackets contribute the 'a' term, whilst the remaining brackets contribute the 'b' term.
Your calculator probably has an nCr button, which makes finding specific terms in large expansions dead easy. No more drawing out massive triangles or doing hundreds of multiplications!
Exam tip: You can find any specific term in a binomial expansion without working out the whole thing - just use nCr for the coefficient you need.
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Mastering the Binomial Expansion
Jess
@jess_eqxp
Binomial expansion is a clever way to expand expressions like (a+b)^n without doing loads of tedious multiplication. It's built on patterns you can spot and techniques that'll save you ages in your A-level maths exams.
Binomial Expansion Basics
Ever wondered how to expand something like ^8 without multiplying it out eight times? Binomial expansion is your answer, and it works for three main types of expressions.
The binomial theorem gives us a shortcut to expand any binomial raised to a power. Instead of multiplying ... repeatedly, we can use patterns to write out the answer directly.
Pascal's triangle is brilliant for finding coefficients. Each row gives you the numbers you need - just start with 1 at the top, then each number below is the sum of the two numbers above it. Row 4 gives you 1, 4, 6, 4, 1 for expanding something to the power of 4.
When expanding ^n, the powers of 'a' start at n and decrease by 1 each term, whilst the powers of 'b' start at 0 and increase by 1 each term. The coefficients come straight from Pascal's triangle.
Quick tip: The row number in Pascal's triangle matches the power you're expanding to!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
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Working Through Examples
Let's see binomial expansion in action with some proper examples that show you exactly how it works.
For ^6, you'll use row 6 of Pascal's triangle: 1, 6, 15, 20, 15, 6, 1. Each term follows the pattern: coefficient × x^(decreasing power) × 2^(increasing power). So you get x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64.
The tricky bit comes with negative terms like ^4. Remember that raised to odd powers stays negative, but even powers become positive. This gives you 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4.
Setting up a table can really help you keep track of all the calculations. List your coefficients, then work out each term systematically - don't try to do it all in your head!
Watch out: When you've got constants multiplying your variables (like the 2 in 2x), you need to raise the entire term to the power, not just the variable.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Binomial Coefficients and Combinations
When you need to expand something massive like ^20, Pascal's triangle becomes impractical. That's where binomial coefficients using combinations come to the rescue.
Factorial notation helps us calculate how many ways we can choose objects. The formula for choosing r objects from n objects is nCr = n!/, and this is exactly what gives us our binomial coefficients.
In any binomial expansion ^n, the coefficient of the term a^r × b^ is nCr. This works because you're essentially choosing which r brackets contribute the 'a' term, whilst the remaining brackets contribute the 'b' term.
Your calculator probably has an nCr button, which makes finding specific terms in large expansions dead easy. No more drawing out massive triangles or doing hundreds of multiplications!
Exam tip: You can find any specific term in a binomial expansion without working out the whole thing - just use nCr for the coefficient you need.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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Students love us — and so will you.
4.9/5
App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user