Tackling polynomial equations just got easier! This guide breaks down... Show more
Pre-Calculus
Algebra 2
Algebra 1
Algebra
Calculus 1
OverigeSign 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
93
•
Dec 17, 2025
•
Understanding Real Zeros in Polynomial Functions
sumehra
@sumehra
1 / 6
Rational Zeros of Polynomials
Ever wonder how to find exactly where a polynomial equals zero? The Rational Zeros Theorem gives us a systematic way to find these special values!
When a polynomial has integer coefficients, any rational zero must be in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. This narrows down our search to just a few possibilities!
For example, with P(x) = x³ - 3x + 2, the possible rational zeros are ±1 and ±2 (since the constant term is 2 and the leading coefficient is 1). Using synthetic division to test these values, we find that 1 and -2 are the actual zeros.
Quick Tip: When the leading coefficient is 1 or -1, your job gets even easier - the rational zeros must be factors of the constant term!
Finding Rational Zeros Step-by-Step
Finding all zeros of a polynomial becomes straightforward with this three-step approach:
- List all possible rational zeros using the Rational Zeros Theorem
- Use synthetic division to test each candidate
- Repeat the process with the resulting quotient until you reach a quadratic expression
Let's see this in action with P(x) = 2x³ + x² - 13x + 6. The possible rational zeros include ±1, ±2, ±3, ±6, ±½, and ±3/2 (factors of 6 divided by factors of 2). Testing x = 2 with synthetic division gives us a zero remainder!
The polynomial factors as , and the quadratic further factors as . This gives us our complete solution: x = ½, -3, and 2.
Remember: Synthetic division is your best friend for testing potential zeros quickly - when you get a remainder of 0, you've found a zero!
Descartes' Rule of Signs
Wouldn't it be great to know how many positive and negative zeros a polynomial has? Descartes' Rule of Signs lets you predict this without solving the equation!
Count the number of sign changes in the coefficients of your polynomial. For P(x) = 5x⁷ - 3x⁵ - x⁴ + 2x² + x - 3, there are 3 sign changes. This means the polynomial has either 3 or 1 positive real zeros.
To find the possible number of negative real zeros, replace x with -x in the original polynomial and count sign changes again. For example, P = 5x⁷ + 3x⁵ - x⁴ + 2x² - x - 3 has 4 sign changes, meaning there are either 4, 2, or 0 negative real zeros.
Math Hack: The actual number of positive or negative zeros will always differ from the number of sign changes by an even number (0, 2, 4, etc.), which narrows down your possibilities significantly!
Upper and Lower Bounds for Roots
Finding where all the zeros of a polynomial are located helps you narrow your search! An upper bound b means all real zeros are less than b, while a lower bound a means all real zeros are greater than a.
You can determine these bounds using synthetic division:
- For an upper bound b (where b > 0): divide by and check if all entries in the result row are non-negative
- For a lower bound a (where a < 0): divide by and check if the signs alternate properly
For instance, with P(x) = x² - 2x + 1, we can determine it has at most 2 positive zeros (from Descartes' Rule) and no negative zeros. This tells us all zeros must be positive.
Visualization Tip: Thinking of bounds as "fences" that contain all the real zeros helps you visualize where to look for solutions on a graph!
Applying Upper and Lower Bounds
Determining where all solutions lie makes solving polynomial equations much easier! Let's find bounds for P(x) = x⁴ - 3x² + 2x - 5.
Using synthetic division with x = 2:
2 | 1 0 -3 2 -5
2 4 2 8
1 2 1 4 3
Since all numbers in the bottom row are positive, 2 is an upper bound.
For x = -3:
-3 | 1 0 -3 2 -5
-3 9 -18 48
1 -3 6 -16 43
The signs alternate properly, so -3 is a lower bound.
This means all real zeros of this polynomial lie between -3 and 2. Knowing this range helps us set up our graphing calculator to find the exact solutions efficiently!
Problem-Solving Strategy: Always check bounds before graphing - it saves time and prevents missing solutions that might lie outside your viewing window!
Solving Polynomial Equations with Technology
Modern graphing technology makes solving complex polynomial equations easier, but you still need algebra to set up your viewing window correctly!
When solving 3x⁴ + 4x³ - 7x² - 2x - 3 = 0:
- First, find upper and lower bounds using synthetic division
- Testing x = 2 and x = -3 confirms these are good bounds
- Set your graphing window to show horizontally and vertically
- Look for where the polynomial graph crosses the x-axis
The bounds tell you exactly where to look, so you won't miss any solutions. Technology handles the calculations while your algebraic knowledge ensures you're looking in the right place!
Real-World Application: Engineers and scientists use these exact techniques to find solutions to complex problems where equations can't be solved by hand!
We thought you’d never ask...
What is the Knowunity AI companion?
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.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the 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: Zeros of Polynomials
Most popular content in Algebra 2
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
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
Understanding Real Zeros in Polynomial Functions
sumehra
@sumehra
Tackling polynomial equations just got easier! This guide breaks down how to find real zeros of polynomials using powerful methods like the Rational Zeros Theorem and Descartes' Rule of Signs. Whether you're spotting patterns in coefficients or finding upper and... Show more
Rational Zeros of Polynomials
Ever wonder how to find exactly where a polynomial equals zero? The Rational Zeros Theorem gives us a systematic way to find these special values!
When a polynomial has integer coefficients, any rational zero must be in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. This narrows down our search to just a few possibilities!
For example, with P(x) = x³ - 3x + 2, the possible rational zeros are ±1 and ±2 (since the constant term is 2 and the leading coefficient is 1). Using synthetic division to test these values, we find that 1 and -2 are the actual zeros.
Quick Tip: When the leading coefficient is 1 or -1, your job gets even easier - the rational zeros must be factors of the constant term!
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
Finding Rational Zeros Step-by-Step
Finding all zeros of a polynomial becomes straightforward with this three-step approach:
- List all possible rational zeros using the Rational Zeros Theorem
- Use synthetic division to test each candidate
- Repeat the process with the resulting quotient until you reach a quadratic expression
Let's see this in action with P(x) = 2x³ + x² - 13x + 6. The possible rational zeros include ±1, ±2, ±3, ±6, ±½, and ±3/2 (factors of 6 divided by factors of 2). Testing x = 2 with synthetic division gives us a zero remainder!
The polynomial factors as , and the quadratic further factors as . This gives us our complete solution: x = ½, -3, and 2.
Remember: Synthetic division is your best friend for testing potential zeros quickly - when you get a remainder of 0, you've found a zero!
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
Descartes' Rule of Signs
Wouldn't it be great to know how many positive and negative zeros a polynomial has? Descartes' Rule of Signs lets you predict this without solving the equation!
Count the number of sign changes in the coefficients of your polynomial. For P(x) = 5x⁷ - 3x⁵ - x⁴ + 2x² + x - 3, there are 3 sign changes. This means the polynomial has either 3 or 1 positive real zeros.
To find the possible number of negative real zeros, replace x with -x in the original polynomial and count sign changes again. For example, P = 5x⁷ + 3x⁵ - x⁴ + 2x² - x - 3 has 4 sign changes, meaning there are either 4, 2, or 0 negative real zeros.
Math Hack: The actual number of positive or negative zeros will always differ from the number of sign changes by an even number (0, 2, 4, etc.), which narrows down your possibilities significantly!
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
Upper and Lower Bounds for Roots
Finding where all the zeros of a polynomial are located helps you narrow your search! An upper bound b means all real zeros are less than b, while a lower bound a means all real zeros are greater than a.
You can determine these bounds using synthetic division:
- For an upper bound b (where b > 0): divide by and check if all entries in the result row are non-negative
- For a lower bound a (where a < 0): divide by and check if the signs alternate properly
For instance, with P(x) = x² - 2x + 1, we can determine it has at most 2 positive zeros (from Descartes' Rule) and no negative zeros. This tells us all zeros must be positive.
Visualization Tip: Thinking of bounds as "fences" that contain all the real zeros helps you visualize where to look for solutions on a graph!
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
Applying Upper and Lower Bounds
Determining where all solutions lie makes solving polynomial equations much easier! Let's find bounds for P(x) = x⁴ - 3x² + 2x - 5.
Using synthetic division with x = 2:
2 | 1 0 -3 2 -5
2 4 2 8
1 2 1 4 3
Since all numbers in the bottom row are positive, 2 is an upper bound.
For x = -3:
-3 | 1 0 -3 2 -5
-3 9 -18 48
1 -3 6 -16 43
The signs alternate properly, so -3 is a lower bound.
This means all real zeros of this polynomial lie between -3 and 2. Knowing this range helps us set up our graphing calculator to find the exact solutions efficiently!
Problem-Solving Strategy: Always check bounds before graphing - it saves time and prevents missing solutions that might lie outside your viewing window!
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 Polynomial Equations with Technology
Modern graphing technology makes solving complex polynomial equations easier, but you still need algebra to set up your viewing window correctly!
When solving 3x⁴ + 4x³ - 7x² - 2x - 3 = 0:
- First, find upper and lower bounds using synthetic division
- Testing x = 2 and x = -3 confirms these are good bounds
- Set your graphing window to show horizontally and vertically
- Look for where the polynomial graph crosses the x-axis
The bounds tell you exactly where to look, so you won't miss any solutions. Technology handles the calculations while your algebraic knowledge ensures you're looking in the right place!
Real-World Application: Engineers and scientists use these exact techniques to find solutions to complex problems where equations can't be solved by hand!
We thought you’d never ask...
What is the Knowunity AI companion?
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.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the 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.
1
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: Zeros of Polynomials
Most popular content in Algebra 2
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
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