Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Pre-Calculus
Nov 29, 2025
197
4 pages
~Taffi~ @taffi
Polynomial functions are math expressions that combine variables and exponents in specific ways. They create smooth curves on... Show more
A polynomial function follows the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where the coefficients are real numbers and n is a positive integer. These functions create smooth curves with no breaks or gaps and are continuous for all real numbers.
The degree of a polynomial is its highest exponent, which determines its classification. For example, degree 0 is constant , degree 1 is linear , degree 2 is quadratic , and degree 3 is cubic .
The end behavior of a polynomial depends on the power and coefficient of its leading term. For even powers with positive coefficients, both ends point upward (↗↗); with negative coefficients, both ends point downward (↘↘). For odd powers with positive coefficients, left end points down and right end points up (↘↗); with negative coefficients, left end points up and right end points down (↗↘).
🔍 Try This Identify the degree and end behavior of f(x) = -2x³+5x-1. Since the highest power is 3 (odd) with a negative coefficient, the function will rise on the left and fall on the right.
Turning points are where polynomial functions change direction from increasing to decreasing (or vice versa). A polynomial of degree n can have at most turning points, which makes this a key feature for identifying polynomial degree.
When a turning point is higher than nearby points, it's called a relative maximum. When it's lower than nearby points, it's called a relative minimum. These maximum and minimum values together are called extrema.
You can find turning points using a graphing calculator by following these steps First, graph the function; then use the CALC menu to find minimum and maximum values; finally, identify the left and right bounds of the turning point and get its coordinates.
💡 Quick Tip For the function f(x) = x³+5x²-8x-1, the relative maximum is at (-4,47) and the relative minimum is at (0.67,-3.48). The function increases from (-∞,-4) and (0.67,∞), while decreasing from (-4,0.67).
Zeros are where a polynomial function equals zero and are also called roots or x-intercepts. A polynomial of degree n can have at most n real zeros. Even-degree polynomials have an even number of real zeros (or none), while odd-degree polynomials always have at least one real zero.
The solutions to the equation f(x) = 0 are the zeros of the function. If x = c is a solution, then is a linear factor of the polynomial. You can find linear factors by factoring the polynomial.
For example, given f(x) = x³ + 5x² - 24x, we can factor it as f(x) = x = x. The linear factors are x, , and , which means the zeros are x = 0, x = -8, and x = 3.
🎯 Remember Every zero of a polynomial corresponds to a factor, and every factor reveals a zero. Knowing how to factor polynomials is essential for finding all zeros quickly.
The multiplicity of a zero is the degree of its corresponding linear factor in the factored form of the polynomial. Multiplicity tells us how a graph behaves at each zero.
When a zero has an even multiplicity , the graph is tangent to the x-axis at that point—it touches but doesn't cross the axis. When a zero has an odd multiplicity , the graph crosses through the x-axis at that point.
For example, in f(x) = x²+10x² + 25x = x², the zeros are x = 0 with multiplicity 1 (odd) and x = -5 with multiplicity 2 (even). At x = 0, the graph crosses the x-axis, while at x = -5, the graph touches the x-axis without crossing.
🔍 Visual Clue When graphing polynomials, look at how the curve behaves at each zero. If it bounces off the x-axis, the multiplicity is even. If it passes through the x-axis, the multiplicity is odd.
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
2
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App 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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
~Taffi~
@taffi
Polynomial functions are math expressions that combine variables and exponents in specific ways. They create smooth curves on graphs with predictable behaviors that depend on their degree and coefficients. Understanding how these functions behave will help you analyze their graphs... Show more
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
A polynomial function follows the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where the coefficients are real numbers and n is a positive integer. These functions create smooth curves with no breaks or gaps and are continuous for all real numbers.
The degree of a polynomial is its highest exponent, which determines its classification. For example, degree 0 is constant , degree 1 is linear , degree 2 is quadratic , and degree 3 is cubic .
The end behavior of a polynomial depends on the power and coefficient of its leading term. For even powers with positive coefficients, both ends point upward (↗↗); with negative coefficients, both ends point downward (↘↘). For odd powers with positive coefficients, left end points down and right end points up (↘↗); with negative coefficients, left end points up and right end points down (↗↘).
🔍 Try This: Identify the degree and end behavior of f(x) = -2x³+5x-1. Since the highest power is 3 (odd) with a negative coefficient, the function will rise on the left and fall on the right.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Turning points are where polynomial functions change direction from increasing to decreasing (or vice versa). A polynomial of degree n can have at most turning points, which makes this a key feature for identifying polynomial degree.
When a turning point is higher than nearby points, it's called a relative maximum. When it's lower than nearby points, it's called a relative minimum. These maximum and minimum values together are called extrema.
You can find turning points using a graphing calculator by following these steps: First, graph the function; then use the CALC menu to find minimum and maximum values; finally, identify the left and right bounds of the turning point and get its coordinates.
💡 Quick Tip: For the function f(x) = x³+5x²-8x-1, the relative maximum is at (-4,47) and the relative minimum is at (0.67,-3.48). The function increases from (-∞,-4) and (0.67,∞), while decreasing from (-4,0.67).
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Zeros are where a polynomial function equals zero and are also called roots or x-intercepts. A polynomial of degree n can have at most n real zeros. Even-degree polynomials have an even number of real zeros (or none), while odd-degree polynomials always have at least one real zero.
The solutions to the equation f(x) = 0 are the zeros of the function. If x = c is a solution, then is a linear factor of the polynomial. You can find linear factors by factoring the polynomial.
For example, given f(x) = x³ + 5x² - 24x, we can factor it as f(x) = x = x. The linear factors are x, , and , which means the zeros are x = 0, x = -8, and x = 3.
🎯 Remember: Every zero of a polynomial corresponds to a factor, and every factor reveals a zero. Knowing how to factor polynomials is essential for finding all zeros quickly.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The multiplicity of a zero is the degree of its corresponding linear factor in the factored form of the polynomial. Multiplicity tells us how a graph behaves at each zero.
When a zero has an even multiplicity , the graph is tangent to the x-axis at that point—it touches but doesn't cross the axis. When a zero has an odd multiplicity , the graph crosses through the x-axis at that point.
For example, in f(x) = x²+10x² + 25x = x², the zeros are x = 0 with multiplicity 1 (odd) and x = -5 with multiplicity 2 (even). At x = 0, the graph crosses the x-axis, while at x = -5, the graph touches the x-axis without crossing.
🔍 Visual Clue: When graphing polynomials, look at how the curve behaves at each zero. If it bounces off the x-axis, the multiplicity is even. If it passes through the x-axis, the multiplicity is odd.
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
2
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App 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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Algebra 2
Calculus 1
Algebra
Algebra 1
Overige