# Polynomial Division

## Practise dividing one algebraic expression by another in this set of exercises.

##### Menu  Level 1Level 2Level 3Level 4Level 5HelpMore Algebra

This is level 5: mixed exercise from old textbook. This long, difficult exercise is only included here as a challenge for the very adventurous. The real challenge is being able to type in the answers so that they exactly match the answers given at the back of the ancient textbook this exercise was taken from (A First Book in Algebra, by Wallace C. Boyden written in 1895)

 1 $$\frac{x^2+8x-105}{x+15}$$ = 2 $$\frac{x^2+8x-33}{x+11}$$ = 3 $$\frac{x^4+x^2-20}{x^2-4}$$ = 4 $$\frac{y^4-y^2-30}{y^2+5}$$ = 5 $$\frac{x^4-31x^2+9}{x^2+5x-3}$$ = 6 $$\frac{a^4-12a^2+16}{a^2-2a-4}$$ = 7 $$\frac{x^3-y^3}{x-y}$$ = 8 $$\frac{a^3+b^3}{a+b}$$ = 9 $$\frac{16a^4-81b^4}{2a-3b}$$ = 10 $$\frac{81x^8-y^4}{3x^2-y}$$ = 11 $$\frac{x^5-x^4y-2x^3y^2-5x^2y^3-17xy^4-12y^5}{x^2-2xy-3y^2}$$ = 12 $$\frac{a^5+a^4b-14a^3b^2+15a^2b^3+7ab^4-10b^5}{a^2-3ab+2b^2}$$ = 13 $$\frac{x^6 - 5x^3 + 3 + 5x^4 - 10x - x^5 + 10x^2}{x^2 + 3 -x}$$ = 14 $$\frac{x^6 - 2x^3 - 2 + x - 3x^5 + 2x^4 - 5x^2}{x^3 + 2 + x}$$ = 15 $$\frac{a^5 - a - 2a^2 - a^3}{a + a^3 + a^2}$$ = 16 $$\frac{x^6 - 2x^3 - x^2 - x^4}{1 + x^2 + x}$$ = 17 $$\frac{a^{11} - a^2}{a^3 - 1}$$ = 18 $$\frac{a^{12} - a^4}{a^2 + 1}$$ = 19 $$\frac{x^4 - x^2y^2 - 2xy^3 + 2y^4}{x^2 - 2xy + y^2}$$ = 20 $$\frac{4a^4 + 81b^4}{2a^2 + 6ab + 9b^2}$$ = 21 $$\frac{\frac{1}{2}x^4 + \frac{3}{4}x^3y - \frac{1}{3}x^2y^2 + \frac{7}{6}xy^3 - \frac{1}{3}y^4}{x^2 - \frac{1}{2}xy + y^2}$$ = 22 $$\frac{\frac{2}{9}y^3 - \frac{5}{36}x^2y + \frac{1}{6}xy^2 + \frac{1}{6}x^3}{ \frac{1}{2}x + \frac{1}{3}y}$$ = 23 $$\frac{\frac{1}{4}(x - y)^5 - (x - y)^3 - \frac{1}{2}(x - y)^2 -\frac{1}{16}(x - y)}{\frac{1}{2}(x - y)^2 + (x - y) +\frac{1}{4}}$$ = 24 $$\frac{16x^8 - 81y^4}{27y^3 + 18x^2y^2 + 8x^6 + 12x^4y}$$ = 25 $$\frac{4x^4 - 10x^2 + 6}{x + 1}$$ = 26 $$\frac{4a^4 - 5a^2b^2 + b^4}{2a - b}$$ = 27 $$\frac{a^6 - b^6 + a^4 + b^4 + a^2b^2}{a^2 - b^2 + 1}$$ = 28 $$\frac{x^6 - y^6 - x^4 - y^4 - x^2y^2}{x^2 - y^2 - 1}$$ = 29 $$\frac{81a^{12} - 16b^8}{12a^3b^4 - 8b^6 - 18a^6b^2 + 27a^9}$$ = 30 $$\frac{-6x^5-12x^4+25x^3-10x^2-14x+12}{-3x^2+2}$$ =
Check

Did you know that there is a method for finding the remainder without having to do long division? It is called:

## The Remainder Theorem

If a polynomial $$f(x)$$ is divided by $$(x-a)$$ then the remainder is $$f(a)$$

Do some independent research to find out more about this theorem and how it can be used to complete this exercise really quickly.

You may be interested to know that students were answering these very same questions over one hundred years ago. This exercise comes from a textbook written in the 1890s.

This is Polynomial Division level 5. You can also try:
Level 1 Level 2 Level 3 Level 4

## Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

## More Activities:

Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

Comment recorded on the 9 May 'Starter of the Day' page by Liz, Kuwait:

"I would like to thank you for the excellent resources which I used every day. My students would often turn up early to tackle the starter of the day as there were stamps for the first 5 finishers. We also had a lot of fun with the fun maths. All in all your resources provoked discussion and the students had a lot of fun."

Comment recorded on the 23 September 'Starter of the Day' page by Judy, Chatsmore CHS:

"This triangle starter is excellent. I have used it with all of my ks3 and ks4 classes and they are all totally focused when counting the triangles."

Each month a newsletter is published containing details of the new additions to the Transum website and a new puzzle of the month.

The newsletter is then duplicated as a podcast which is available on the major delivery networks. You can listen to the podcast while you are commuting, exercising or relaxing.

Transum breaking news is available on Twitter @Transum and if that's not enough there is also a Transum Facebook page.

#### Without Lifting

Can you draw these diagrams without lifting your pencil from the paper? This is an interactive version of the traditional puzzle. Some diagrams are possible while others are not. What is the rule?

There are answers to this exercise but they are available in this space to teachers, tutors and parents who have logged in to their Transum subscription on this computer.

A Transum subscription unlocks the answers to the online exercises, quizzes and puzzles. It also provides the teacher with access to quality external links on each of the Transum Topic pages and the facility to add to the collection themselves.

Subscribers can manage class lists, lesson plans and assessment data in the Class Admin application and have access to reports of the Transum Trophies earned by class members.

Subscribe

## Go Maths

Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school.

## Maths Map

Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.

## Teachers

If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows:

Alternatively, if you use Google Classroom, all you have to do is click on the green icon below in order to add this activity to one of your classes.

It may be worth remembering that if Transum.org should go offline for whatever reason, there is a mirror site at Transum.info that contains most of the resources that are available here on Transum.org.

When planning to use technology in your lesson always have a plan B!

Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.

For Students:

For All:

Scan the QR code below to visit the online version of this activity.

https://www.Transum.org/go/?Num=904

## Description of Levels

Close

Algebraic Fractions - For a simpler introduction have a go at Algebraic Fractions level 1.

Level 1 - Divide a polynomial by a single term

Level 2 - Divide a polynomial by a linear expression

Level 3 - Find the remainder

Level 4 - Divide a polynomial by a quadratic or cubic expression

Level 5 - Mixed exercise from old textbook

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More Algebra including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

## Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

## Divident ÷ Divisor = Quotient

To type indices or exponents use the up arrow key ^ then type the number followed by the right arrow. The terms in your answer should be in the same order as the terms in the question.

## Beth's Method (see video above)

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

### Typing Mathematical Notation

These exercises use MathQuill, a web formula editor designed to make typing Maths easy and beautiful. Watch the animation below to see how common mathematical notation can be created using your keyboard.

Close