Ever wondered how to tackle those tricky cubic and quartic... Show more
SociologySign 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
3
0
K
Kate Robinson
02/12/2025
Further Maths
Algebraic Division, Factor and Remainder Theorem
208
•
2 Dec 2025
•
Understanding Algebraic Division: Master the Factor and Remainder Theorem
K
Kate Robinson
@katerobinson_rafs
1 / 5
Algebraic Division Basics
Think of algebraic division like long division with numbers, but using polynomials instead. It's absolutely essential for solving cubic equations and higher-order polynomials that would otherwise be nearly impossible to tackle.
The process follows the same pattern as numerical long division. You divide the highest power terms first, multiply back, subtract, then bring down the next term. Keep your expressions in descending order (highest powers first) and use zeros for missing terms to keep everything aligned.
When you get a remainder of zero, you've found a factor. This is brilliant news because it means you can break your complex polynomial into simpler parts that are much easier to work with.
Quick Tip: Always arrange terms in descending order of powers and fill in missing terms with zeros - this keeps your work tidy and prevents mistakes!
The Factor Theorem
Here's where maths gets really clever: the factor theorem connects factors with solutions in a beautifully simple way. If is a factor of polynomial f(x), then f(a) = 0. It works both ways too - if f(a) = 0, then must be a factor.
This theorem is incredibly useful because you can test potential factors quickly without doing lengthy division. Just substitute the value and see if you get zero. If you do, you've found a factor!
Once you've found one factor using the theorem, you can use algebraic division to find the remaining factors. This transforms a complex cubic equation into simpler quadratic factors that you can solve using familiar methods.
Memory Hack: Think "zero in, factor out" - if substituting a value gives zero, that value gives you a factor!
Solving Cubic Equations
Solving cubic equations becomes much more manageable when you combine the factor theorem with algebraic division. Start by testing simple values like ±1, ±2, ±3 to find your first factor quickly.
Once you've found that first factor , divide your cubic by this factor using algebraic division. You'll end up with a quadratic expression that you can factorise using standard methods or the quadratic formula.
The beauty of this approach is that it breaks down what seems like an impossible problem into steps you already know how to handle. A cubic equation with three solutions becomes much less intimidating when you tackle it systematically.
Pro Strategy: Always check your factorisation by expanding it back out - if you get your original expression, you know you're on the right track!
The Remainder Theorem
The remainder theorem is like the factor theorem's cousin - it tells you what's left over when you divide. For polynomial f(x), the value f equals the remainder when f(x) is divided by .
This theorem is incredibly handy because it means you can find remainders without actually doing the division. Just substitute the appropriate value into your polynomial and you've got your answer instantly.
You'll often see exam questions where they give you the remainder and ask you to find an unknown coefficient. Simply set up the equation using the remainder theorem and solve for the unknown.
Time Saver: Use the remainder theorem instead of long division whenever possible - it's much quicker and less prone to arithmetic errors!
Advanced Applications
When exam questions combine polynomial division with unknown coefficients, don't panic - just work systematically. Use the remainder theorem to set up equations, then solve for the unknowns step by step.
The factor theorem becomes particularly powerful when you need to factorise polynomials completely. Test values methodically until you find factors, then use division to reduce the problem to something simpler.
For equations involving polynomial expressions set equal to other polynomials, remember that you can rearrange and factorise. Look for common factors and use the zero product property to find all solutions, including repeated roots.
Exam Success: These theorems often appear together in exam questions - master the connections between them and you'll handle even complex polynomial problems with confidence!
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.
Most popular content: Remainder Theorem
Most popular content in Further Maths
Most popular content
Can't find what you're looking for? Explore other subjects.
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
Understanding Algebraic Division: Master the Factor and Remainder Theorem
K
Kate Robinson
@katerobinson_rafs
Ever wondered how to tackle those tricky cubic and quartic equations that seem impossible to solve? Algebraic division is your secret weapon for breaking down complex polynomials into manageable pieces, making even the most intimidating higher-order equations solvable.
Algebraic Division Basics
Think of algebraic division like long division with numbers, but using polynomials instead. It's absolutely essential for solving cubic equations and higher-order polynomials that would otherwise be nearly impossible to tackle.
The process follows the same pattern as numerical long division. You divide the highest power terms first, multiply back, subtract, then bring down the next term. Keep your expressions in descending order (highest powers first) and use zeros for missing terms to keep everything aligned.
When you get a remainder of zero, you've found a factor. This is brilliant news because it means you can break your complex polynomial into simpler parts that are much easier to work with.
Quick Tip: Always arrange terms in descending order of powers and fill in missing terms with zeros - this keeps your work tidy and prevents mistakes!
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
The Factor Theorem
Here's where maths gets really clever: the factor theorem connects factors with solutions in a beautifully simple way. If is a factor of polynomial f(x), then f(a) = 0. It works both ways too - if f(a) = 0, then must be a factor.
This theorem is incredibly useful because you can test potential factors quickly without doing lengthy division. Just substitute the value and see if you get zero. If you do, you've found a factor!
Once you've found one factor using the theorem, you can use algebraic division to find the remaining factors. This transforms a complex cubic equation into simpler quadratic factors that you can solve using familiar methods.
Memory Hack: Think "zero in, factor out" - if substituting a value gives zero, that value gives you a factor!
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
Solving Cubic Equations
Solving cubic equations becomes much more manageable when you combine the factor theorem with algebraic division. Start by testing simple values like ±1, ±2, ±3 to find your first factor quickly.
Once you've found that first factor , divide your cubic by this factor using algebraic division. You'll end up with a quadratic expression that you can factorise using standard methods or the quadratic formula.
The beauty of this approach is that it breaks down what seems like an impossible problem into steps you already know how to handle. A cubic equation with three solutions becomes much less intimidating when you tackle it systematically.
Pro Strategy: Always check your factorisation by expanding it back out - if you get your original expression, you know you're on the right track!
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
The Remainder Theorem
The remainder theorem is like the factor theorem's cousin - it tells you what's left over when you divide. For polynomial f(x), the value f equals the remainder when f(x) is divided by .
This theorem is incredibly handy because it means you can find remainders without actually doing the division. Just substitute the appropriate value into your polynomial and you've got your answer instantly.
You'll often see exam questions where they give you the remainder and ask you to find an unknown coefficient. Simply set up the equation using the remainder theorem and solve for the unknown.
Time Saver: Use the remainder theorem instead of long division whenever possible - it's much quicker and less prone to arithmetic errors!
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
Advanced Applications
When exam questions combine polynomial division with unknown coefficients, don't panic - just work systematically. Use the remainder theorem to set up equations, then solve for the unknowns step by step.
The factor theorem becomes particularly powerful when you need to factorise polynomials completely. Test values methodically until you find factors, then use division to reduce the problem to something simpler.
For equations involving polynomial expressions set equal to other polynomials, remember that you can rearrange and factorise. Look for common factors and use the zero product property to find all solutions, including repeated roots.
Exam Success: These theorems often appear together in exam questions - master the connections between them and you'll handle even complex polynomial problems with confidence!
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.
3
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Mock Exam
Quiz
Flashcards
Essay
Most popular content: Remainder Theorem
Most popular content in Further Maths
Most popular content
Can't find what you're looking for? Explore other subjects.
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