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Maths
25 Nov 2025
115
7 pages
Emily Kelt @emilykelt_yrng
This comprehensive guide covers polynomial division, remainder theorem, and completing the square techniques. The material progresses from basic... Show more
Page 2 Advanced Polynomial Division
This section expands on synthetic division techniques with more complex examples involving common factors and multiple terms.
Example Division of demonstrating how to handle common factors.
Vocabulary Common factor - a term that divides evenly into multiple terms of a polynomial.
Highlight Special attention is given to cases where common factors must be considered before synthetic division.
Page 3 Finding Polynomial Roots
This page focuses on techniques for finding polynomial roots using factorization and the remainder theorem.
Definition A root of a polynomial is a value that makes the polynomial equal to zero.
Example Detailed solution of showing how to find roots systematically.
Highlight The discriminant method is introduced for cases where factorization is not possible.
Page 4 Unknown Coefficients
This section deals with determining unknown coefficients in polynomials using synthetic division and simultaneous equations.
Example Solution of a polynomial with unknown coefficients a and b, demonstrating how to set up and solve simultaneous equations.
Highlight The importance of using synthetic division to verify solutions and find additional roots.
Page 5 Completing the Square
Detailed explanation of completing the square technique for quadratic expressions.
Definition Completing the square transforms a quadratic expression into the form ² + b.
Example Transformation of x² + 6x + 3 into completed square form.
Highlight Special attention to identifying maximum and minimum points from completed square form.
Page 6 The Discriminant
Comprehensive coverage of the discriminant formula and its applications in determining the nature of roots.
Definition The discriminant b² - 4ac determines the nature of quadratic roots.
Example Analysis of x² + 5x - 7 using the discriminant.
Highlight The relationship between discriminant values and the nature of roots (real, equal, or no real roots).
Page 7 Advanced Root Analysis
This final page covers advanced applications of discriminant analysis for determining root characteristics.
Example Analysis of x² + x + m to determine values of m for distinct roots.
Highlight The importance of graphical interpretation in understanding root behavior.
Page 1 Synthetic Division and Remainder Theorem
This page introduces fundamental concepts of polynomial synthetic division and the remainder theorem. The content demonstrates how to perform synthetic division with detailed step-by-step examples.
Definition Synthetic division is a shorthand method for dividing polynomials where the coefficients of the polynomial are used in a tabular arrangement.
Example Division of by , showing the complete process including remainder calculation.
Highlight The remainder theorem states that the remainder of a polynomial p(x) divided by equals p(a).
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.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
8
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Explore key techniques for solving simultaneous equations, including quadratic equations and linear systems. This resource covers various methods, such as substitution and elimination, with detailed examples and solutions. Ideal for Edexcel A Level Maths students preparing for exams.
MathsExplore essential mathematical concepts including polynomial theorems, logarithmic properties, trigonometric functions, and integration techniques. This resource covers everything from solving inequalities to understanding exponential functions, providing a solid foundation for A-level mathematics. Ideal for students aiming for top grades.
MathsExplore a comprehensive set of Edexcel A-Level Maths inequalities problems, including compound inequalities, linear inequalities, and multi-step solutions. This resource provides detailed examples and solutions to enhance your understanding of solving inequalities and graphing techniques.
MathsExplore the detailed solution for optimizing the surface area of a cuboid box with a tunnel, as presented in the SQA Higher Mathematics 2019 exam. This study note covers the derivation of the surface area formula, the application of differentiation to find minimum values, and step-by-step explanations for both parts A and B of the question. Ideal for students preparing for optimization problems in mathematics.
MathsExplore the three main types of mathematical proofs: Proof by Exhaustion, Proof by Contradiction, and Proof by Deduction. This summary includes examples related to prime numbers, square numbers, and geometric proofs, providing a clear understanding of logical reasoning in mathematics.
MathsExplore key concepts in solving quadratic equations, including the Quadratic Formula, completing the square, and understanding the discriminant. This summary provides essential examples and explanations for mastering quadratic functions and their graphs.
MathsApp Store
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
Emily Kelt
@emilykelt_yrng
This comprehensive guide covers polynomial division, remainder theorem, and completing the square techniques. The material progresses from basic concepts to complex problem-solving applications, making it an essential resource for students studying advanced algebra.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Page 2: Advanced Polynomial Division
This section expands on synthetic division techniques with more complex examples involving common factors and multiple terms.
Example: Division of demonstrating how to handle common factors.
Vocabulary: Common factor - a term that divides evenly into multiple terms of a polynomial.
Highlight: Special attention is given to cases where common factors must be considered before synthetic division.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Page 3: Finding Polynomial Roots
This page focuses on techniques for finding polynomial roots using factorization and the remainder theorem.
Definition: A root of a polynomial is a value that makes the polynomial equal to zero.
Example: Detailed solution of showing how to find roots systematically.
Highlight: The discriminant method is introduced for cases where factorization is not possible.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Page 4: Unknown Coefficients
This section deals with determining unknown coefficients in polynomials using synthetic division and simultaneous equations.
Example: Solution of a polynomial with unknown coefficients a and b, demonstrating how to set up and solve simultaneous equations.
Highlight: The importance of using synthetic division to verify solutions and find additional roots.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Page 5: Completing the Square
Detailed explanation of completing the square technique for quadratic expressions.
Definition: Completing the square transforms a quadratic expression into the form ² + b.
Example: Transformation of x² + 6x + 3 into completed square form.
Highlight: Special attention to identifying maximum and minimum points from completed square form.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Page 6: The Discriminant
Comprehensive coverage of the discriminant formula and its applications in determining the nature of roots.
Definition: The discriminant b² - 4ac determines the nature of quadratic roots.
Example: Analysis of x² + 5x - 7 using the discriminant.
Highlight: The relationship between discriminant values and the nature of roots (real, equal, or no real roots).
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Page 7: Advanced Root Analysis
This final page covers advanced applications of discriminant analysis for determining root characteristics.
Example: Analysis of x² + x + m to determine values of m for distinct roots.
Highlight: The importance of graphical interpretation in understanding root behavior.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Page 1: Synthetic Division and Remainder Theorem
This page introduces fundamental concepts of polynomial synthetic division and the remainder theorem. The content demonstrates how to perform synthetic division with detailed step-by-step examples.
Definition: Synthetic division is a shorthand method for dividing polynomials where the coefficients of the polynomial are used in a tabular arrangement.
Example: Division of by , showing the complete process including remainder calculation.
Highlight: The remainder theorem states that the remainder of a polynomial p(x) divided by equals p(a).
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.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
8
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Explore key techniques for solving simultaneous equations, including quadratic equations and linear systems. This resource covers various methods, such as substitution and elimination, with detailed examples and solutions. Ideal for Edexcel A Level Maths students preparing for exams.
MathsExplore essential mathematical concepts including polynomial theorems, logarithmic properties, trigonometric functions, and integration techniques. This resource covers everything from solving inequalities to understanding exponential functions, providing a solid foundation for A-level mathematics. Ideal for students aiming for top grades.
MathsExplore a comprehensive set of Edexcel A-Level Maths inequalities problems, including compound inequalities, linear inequalities, and multi-step solutions. This resource provides detailed examples and solutions to enhance your understanding of solving inequalities and graphing techniques.
MathsExplore the detailed solution for optimizing the surface area of a cuboid box with a tunnel, as presented in the SQA Higher Mathematics 2019 exam. This study note covers the derivation of the surface area formula, the application of differentiation to find minimum values, and step-by-step explanations for both parts A and B of the question. Ideal for students preparing for optimization problems in mathematics.
MathsExplore the three main types of mathematical proofs: Proof by Exhaustion, Proof by Contradiction, and Proof by Deduction. This summary includes examples related to prime numbers, square numbers, and geometric proofs, providing a clear understanding of logical reasoning in mathematics.
MathsExplore key concepts in solving quadratic equations, including the Quadratic Formula, completing the square, and understanding the discriminant. This summary provides essential examples and explanations for mastering quadratic functions and their graphs.
MathsApp Store
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
